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ACTA UNIVERSITATIS UPSALIENSIS UPPSALA 2019 Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology 1836 Mechanical analyses of trabecular bone and its interaction with implants DAN WU ISSN 1651-6214 ISBN 978-91-513-0715-2 urn:nbn:se:uu:diva-385143

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Page 1: Mechanical analyses of trabecular bone and its interaction ...uu.diva-portal.org/smash/get/diva2:1342155/FULLTEXT01.pdf · Since one of the key mechanical properties of trabecular

ACTAUNIVERSITATIS

UPSALIENSISUPPSALA

2019

Digital Comprehensive Summaries of Uppsala Dissertationsfrom the Faculty of Science and Technology 1836

Mechanical analyses of trabecularbone and its interaction withimplants

DAN WU

ISSN 1651-6214ISBN 978-91-513-0715-2urn:nbn:se:uu:diva-385143

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Dissertation presented at Uppsala University to be publicly examined in Ångström 8001,Lägerhyddsvägen 1, Uppsala, Wednesday, 18 September 2019 at 13:15 for the degree ofDoctor of Philosophy. The examination will be conducted in English. Faculty examiner:Professor Hanna Isaksson (Department of Biomedical Engineering, Lund University,Sweden).

AbstractWu, D. 2019. Mechanical analyses of trabecular bone and its interaction with implants.Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science andTechnology 1836. 65 pp. Uppsala: Acta Universitatis Upsaliensis. ISBN 978-91-513-0715-2.

Bone substitute materials or implants are commonly used in the surgical treatment of bonefractures. However, severe complications are sometimes reported. In order to improve fracturetreatment where the interior, porous trabecular bone is involved, it is important to betterunderstand the mechanical properties of this bone and how it interacts with the substitutes/implants, and this was the aim of this thesis.

Since one of the key mechanical properties of trabecular bone, i.e. the elastic moduli at thetissue level, was not consistently reported in the literature, the results from four widely appliedmethods were first summarized and presented in a review paper.

Furthermore, to improve the analysis of the mechanical behavior of bone and its interactionwith implants, a new digital volume correlation (DVC) technique was proposed based on higher-order finite elements.

We further proposed a method to estimate the elastic modulus at the tissue level bycompression of single trabeculae within a synchrotron radiation micro-computed tomograph(SRµCT). Full-field displacements were estimated by DVC, which also provided boundaryconditions for a finite element model. The proposed method shows potential to estimatetrabecular mechanical properties at the tissue level.

Further, strains and cracks of a trabecular structure under compression till fracture werecharacterized at the single trabecular level, with DVC applied on high-resolution images fromSRµCT.

The effect of augmentation materials on the engagement of screws inserted into trabecularbone was evaluated in human femoral bone, with and without real-time SRµCT. A newlydeveloped tissue adhesive indicated a potential benefit of this material to the primary implantstability compared to a cement and no augmentation.

Finally, a trabecular structure of PLA/HA composite material was printed using a fuseddeposition modelling method as a preliminary step towards better synthetic models of trabecularbone. The synthetic trabecular structure was evaluated using micro-CT, compression and screwpull-out testing.

In conclusion, methods to estimate strains and mechanical properties of trabecular bone wereproposed, insights into interactions between trabecular bone and augmentation/implants weregained, as well as a first step towards a synthetic trabecular model, which may contribute tofurther mechanical analyses and/or improved clinical treatments of trabecular bone.

Keywords: Trabecular bone, Elastic modulus, Digital volume correlation, Pullout resistance,Micro-computed tomography.

Dan Wu, Department of Engineering Sciences, Applied Materials Sciences, Box 534, UppsalaUniversity, SE-75121 Uppsala, Sweden.

© Dan Wu 2019

ISSN 1651-6214ISBN 978-91-513-0715-2urn:nbn:se:uu:diva-385143 (http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-385143)

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To my family

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List of Papers

This thesis is based on the following papers, which are referred to in the textby their Roman numerals.

I Wu, D., Isaksson, P., Ferguson, S.J., Persson, C. (2018).Young’s modulus of trabecular bone at the tissue level: A re-view. Acta Biomaterialia, 78:1-12

II van Dijk, N.P., Wu, D., Persson, C., Isaksson, P. (2019). Aglobal digital volume correlation algorithm based on higher-order finite elements: Implementation and evaluation. Interna-tional Journal of Solids and Structures, 168:211-227

III Wu, D., van Dijk, N.P., Joffre, T., Öhman Mägi, C., Ferguson,S.J., Persson, C., Isaksson, P. (2019). A combined experimentaland numerical method to estimate the elastic modulus of singletrabeculae. (Submitted)

IV Wu, D., Isaksson, P., Persson, C. (2019). Quantification ofstrains and cracks in trabecular bone by digital volume correla-tion: A case study. (Submitted)

V Wu, D., Pujari-Palmer, M., Bojan, A., Palmquist, A., ÖhmanMägi, C., Ferguson, S.J., Isaksson, P., Persson, C. (2019). Theeffect of two augmentation materials on screw pullout re-sistance from human trabecular bone. (Submitted)

VI Wu, D., Spanou, A., Diez-Escudero, A., Persson, C. (2019).3D-printed PLA/HA composite structures as synthetic trabecu-lar bone: a feasibility study using Fused Deposition Modelling.(Submitted)

Reprints of Paper I were made with permission from the respective publish-ers. Paper II is published with open access.

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Author’s contribution

I Major part of planning, data analyses and writing.

II Part of planning, evaluation and analyses.

III Major part of planning, experimental work, analyses and writ-ing.

IV Major part of planning, experimental work, analyses and writ-ing.

V Major part of planning, experimental work, analyses and writ-ing.

VI Part of planning, experimental work and major part of writing.

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Contents

1. Introduction ............................................................................................... 11

2. Background ............................................................................................... 132.1 Bone ................................................................................................... 132.2 Trabecular bone .................................................................................. 162.3 Screw implants ................................................................................... 182.4 Imaging and mechanical testing ......................................................... 19

3. Summary of aims and objectives .............................................................. 22

4. Methods .................................................................................................... 234.1 Mechanical testing .............................................................................. 234.2 Micro-computed tomography ............................................................. 244.3 In-situ mechanical testing within SRμCT .......................................... 264.4 Digital volume correlation.................................................................. 27

5. Results and discussion .............................................................................. 30

Paper I .......................................................................................................... 31

Paper II ......................................................................................................... 34

Paper III ....................................................................................................... 38

Paper IV ....................................................................................................... 42

Paper V ........................................................................................................ 44

Paper VI ....................................................................................................... 48

6. Summary and conclusions ........................................................................ 49

7. Future perspectives ................................................................................... 51

Svensk sammanfattning ................................................................................ 52

Acknowledgements ....................................................................................... 54

References ..................................................................................................... 56

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Abbreviations

BV/TV

CPCDVCFDMFEFFTHAHR-pQCT

Micro-CT

Bone volume fraction (bone volume /total volume)Calcium phosphate cementDigital volume correlationFused deposition modellingFinite elementFast-Fourier transformationHydroxyapatiteHigh resolution peripheral quantita-tive computed tomographyMicro-computed tomography

PLAPMMASRµCT

Tb.SpTb.Th

Polylactic acidPolymethylmethacrylateSynchrotron radiation based X-raymicro-computed tomographyTrabecular spacingTrabecular thickness

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1. Introduction

Bone fractures are common in daily life. The incidence of new bone failuresis reported to be 2.68 million per year in six European countries [1] (France,Germany, Italy, Spain, Sweden and the UK). Trauma due to, for instance,falls and accidents, and bone disease are two common causes for bone frac-tures. The prevalence of osteoporosis, causing a majority of fractures, isreported to be 7 to 19% for men and 23 to 35% for women over 50 years old[1, 2], and the number of individuals exceeding the fracture threshold is ex-pected to be more than 300 million worldwide by 2040 [3]. With the agingpopulation and increasing life span, fragility fractures are becoming a barrierto mobility, independence and life quality, especially for the elderly.

In the surgical treatment of bone fractures, bone substitutes or implantsare utilized for various reasons. For instance, to treat bone fractures causedby high-impact trauma, internal implants such as plates and screws arecommonly used to hold together the bone fragments so that the bone canheal by itself. In the treatment of vertebral compression fractures, bone ce-ments, typically acrylic or calcium phosphate cements, have been injected tothe vertebral body for stabilization and pain relief [4], i.e. vertebroplasty.

However, complications may arise, for various reasons. In one of thetreatments aforementioned, for example, a substantial number of patientsreceiving vertebroplasty caused by osteoporosis develop new fractures, es-pecially in vertebrae adjacent to the treated level (Fig. 1) [5, 6]. One possiblereason is that the injected cement alters the rigidity of the vertebra and loadsto the adjacent vertebrae [7, 8]. Although numerous efforts have been put indeveloping better bone substituent materials, it is of the same significance tounderstand the mechanical behavior of bone itself, especially when the re-ported properties vary substantially [9, 10]. Screw implant loosening can beanother frequent complication, as in cases of humeral heads (Fig. 1) [11],osteoporotic vertebrae [12] and skeletal tumors [13], resulting in revision, ora second operation. One important reason for loosening is the inadequateinitial fixation of the screw, which leads to micro-motion of the implants.

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Figure 1. (Left) Scan taken immediately after cement augmentation into multi-levelvertebrae. (Middle) Failure of the adjacent vertebra after approximately 2 weeks.Reprinted with permission from Baroud et al.[14]. (Right) Loosening of pediclescrew indicated by the clear zone. Reprinted with permission from El Saman etal.[15].

Mechanical analyses of bone and its interaction with implants are there-fore fundamental towards improvements in the treatment of bone fractures.A powerful tool in the mechanical analyses of bone [16], bone-cement [17],or bone-implant systems [18, 19], is finite element modeling, especially fortrabecular bone where the naturally complex structures complicate the anal-yses. Additionally, finite element models consume fewer bone samples com-paring to experimental tests. In a future perspective, finite element modelingmay be increasingly used in the analyses of bone through statistical modelsand patient-specific fracture treatment, from lab to clinical applications [20,21]. Crucial inputs for the models to be able to predict physical phenomenaof bone with accuracy are the bone’s mechanical properties at the tissue lev-el.

In this thesis, several aspects in the mechanics of trabecular bone and bone-implant interactions were investigated, with a focus on the elastic modulus of the trabecular bone at tissue level, strain concentrations in the trabecular bone, pullout resistance and failure mechanisms of screws from the trabecular bone with and without augmentation, and 3D-printing of syn-thetic trabecular bone. An algorithm with high accuracy was also proposed to quantitatively estimate the internal displace-ment and strain fields of the tested specimens in the mechanical tests.

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2. Background

2.1 BoneBones, forming the skeletal system, have multiple functions for human be-ings. They support the weight of the body, protect vital organs, and allowour locomotion. Bones work as storage for minerals, and bone marrow with-in the bone continuously supply blood cells [22]. Generally, a bone is com-posed of a hard and dense outer layer, i.e. cortical bone, and more porousinterior, which is called trabecular bone or cancellous bone (Fig. 2). Thecombination of solid cortical and porous trabecular bone provides mechani-cal strength with relatively low density.

Figure 2.Illustration of cancellous and cortical bone with a longitudinal section of the femur. Reprinted with permission from Cowin [23].

In the material perspective, bone tissue consists of 70 wt% inorganic mat-ter, 20 wt% organic and 10 wt% water [24]. The inorganic phase is an im-pure calcium phosphate salt in the form of a modified hydroxyapatite(Ca5(PO4)3OH). About 90 wt% of the organic phase is a type I collagen. It iscommonly deemed that the minerals mainly contribute to the elastic proper-ties and the collagen to the non-linear properties. Any pathology or treatmentthat modifies the three components or their ratio might affect the mechanicalproperties of the bone. Cortical bone exposed to high doses of X-rays thatdestroy the collagen structure showed an adverse effect on strength and frac-ture toughness [25]. Dehydrated bone is reported to be stiffer and more brit-tle [26, 27].

Bone is a hierarchical composite material from the nanoscale to the mac-roscale (Fig. 3) [28, 29]. At the nanoscale, the minerals aggregate and form afractal-like organization [28]. The mineralized collagen fibrils assemble into

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ordered arrays with a diameter from less than one micron to several microns.The arrays then form bundles with different patterns. Sheets of bundles stackin a ply-wood manner, but with the angular offset between adjacent bundlesheets being 40~80°, thus forming lamellar bone. The lamellar bone is foundin both trabecular bone with a motif of lamellar packets and cortical bone inthe form of osteons. A lamellar packet is an assembly of lamellae withslightly different orientations. In an osteon, the lamellae are aligned intolarge concentric rings, similar to that of wood rings. It should be noted thatdisordered fibril arrays and disordered phases also present in the hierarchicallevels of bone.

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Figure 3. Hierarchical levels of bone with representative images, specifically: levelI: Components, II: Structural components, III: Arrays, IV: Array patterns, V: Super-structure, VI: Material patterns, VII: Tissue elements, VIII: Tissue and IX: Organ.The schematic is colored with green representing ordered materials, blue for thedisordered material. The borders of the images are color coded in the same way.Additionally, a graded color represents the presence of both materials. Reprintedwith permission from Reznikov et al. [29].

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2.2 Trabecular boneAt the highest hierarchical level (Fig. 3), also referred to as the apparent level in this thesis, trabecular bone appears as a porous, cellulous structure, containing a network of trabeculae in the shape of small rods and plates (Fig. 4). Typically, the individual trabeculae have diameters between 100 to 300 μm, with a spacing between trabeculae of around 500 to 1500 μm [24]. The term ‘architecture’ or ‘morphology’ is commonly used to refer to the spatial topology or shape of the trabeculae. Trabecular morphology is continuously changing in vivo due to bone remodeling, a process where ma-ture bone matrix is removed and new one is formed. Bone remodeling re-sponds to mechanical loading as expressed by Wolff’s law: if the loading in the trabecular bone increased, the bone morphology will be remodeled to resist the load. The rate of remodeling is also affected by factors such as age and pathology. As a result, the trabecular morphology varies among individ-uals, anatomic sites, age groups, health states etc.

Figure 4. 3D renderings of trabecular bone from human femurs with (left) relativelylow and (right) high bone volume fraction, showing the rod and plate like trabecu-lae.

Due to the close relation between morphology and mechanical behavior incellulous materials, the trabecular morphology needs to be characterizedbefore further analyses of the mechanical properties. Widely used parame-ters include bone volume fraction (BV/TV, bone volume over total volume),trabecular thickness, trabecular spacing, surface-to-volume ratio, connectivi-ty, degree of anisotropy etc. With the application of imaging techniquesproviding 3-dimensional bone structures, for instance, micro-computed to-mography (micro-CT), these morphometric parameters are then calculatedfrom the volumetric digital images. Detailed definitions and calculations ofsome of these parameters are discussed in section 4.2

Many studies have reported correlations between morphometric parame-ters and apparent-level mechanical properties of trabecular bone, such as,stiffness [30], ultimate compression strength [31], and yield strength [32].The mechanical properties are reported to be mainly dependent on theBV/TV, which could explain about 70~89% of variations of the mechanical

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properties. Furthermore, BV/TV together with the fabric tensor, i.e. a secondrank tensor that characterizes the anisotropic architecture of the trabecularbone, have been reported to improve the prediction accuracy to 90~98% [30-32].

It should be noted that the apparent-level mechanical behavior of the tra-becular bone depends not only on the aforementioned morphology, but also on the mechanical properties of the composing individual trabeculae, which are referred to as tissue-level properties. Mechanical tests have been per-formed at different levels to estimate the tissue-level mechanical properties of trabeculae (Fig. 5), from nanoindentation, directly on individual trabec-ulae, to a combination of experiments and finite element simulations on tra-becular structures (5~10 mm). Due to limitations of each method as well as the natural variation between subjects, the reported mechanical properties are in a wide range (Table 1), indicating a demand for further studies to pin-point these properties.

Figure 5. Methods to estimate tissue-level mechanical properties of trabecular bone.Reprinted with permission from Paper I [33].

Table 1. Mechanical properties of trabecular bone at the macroscopic and at thetissue level.

Ultimate compressivestrength [MPa]

Yield strain[%]

Elastic modulus[GPa]

Trabecularstructure 1-30 [34, 35] 0.6-1.2 [34, 36] 0.01-1 [34, 37]

Singletrabeculae 167-208 [38] 0.7-1.4 [38] 1.4-16 [38-40]

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2.3 Screw implantsTwo types of screw implants are widely used in clinics: cortical and cancel-lous screws, as shown in Fig. 6. Cortical screws are mostly used in the di-aphysis of long bones containing mainly cortical bone. The threads of corti-cal screws are shallow and densely spaced to improve the engagement withthe cortical walls. Cancellous screws are used for epiphyses containing sub-stantial amounts of trabecular bone, and the threads are deeper and spacedmore widely to suit the cellulous structure of the trabecular bone.

The thread pitch and depth* of the commonly used cancellous screws (Ta-ble 2) are at the same scale as the trabecular thickness and spacing, whichmeans that trabecular bone around the screws cannot be considered as a con-tinuum [41], which has also been confirmed via finite element analyses [42].Studies have shown that the morphology of the trabecular bone closest to thescrew, especially BV/TV, plays a key role in the pullout stiffness of thescrew engagement [18].

Figure 6. Orthopaedic screws used in the fracture treatment at (A) femur, (B) humerus, and (C) diaphysis of the humerus. Reprinted with permission from Zdero et al. [43].

Table 2. Dimensions of some typical orthopaedic screws corresponding to the ISOstandard 5835:1991.

Outer diameter[mm]

Core diameter[mm]

Pitch[mm]

Trabecular screw 6.5 3.0 2.754.0 1.9 1.75

Cortical screw 4.5 3.0 1.753.5 2.4 1.25

* Thread pitch: the distance between the crests of two adjacent threads along the length of thescrew; Thread depth: the distance between the crest and the base of a thread along the radialdirection.

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In cases where screw implants are inserted into low quality bone, bonecement is sometimes injected to improve the holding strength of the screws,as shown in Fig. 7. So far, the most studied biomaterials for augmentationare polymethylmethacrylate (PMMA) and calcium phosphate cements(CPCs) [44-56]. The augmentation with PMMA generally improves thepullout strength, for instance, from 1.8 to 2.8-fold in vertebrae using pediclescrews [49, 54-56]. However, the improvement of the augmentation withCPCs varies, from apparent reduction under certain conditions [57] to lessimprovement than PMMA [47, 49]. Considering the potential of CPCs interms of biodegradation and osseointegration [45, 58], effort has been putinto improving the understanding of the mechanisms that affect the engage-ment of the bone-screw-cement system [17, 49, 59], aiming at solutions formore consistent improvement in mechanical engagement.

Figure 7. Radiograph of a patient receiving plate and screws in the treatment oftibia fracture. Calcium phosphate cement is used around the long trabecular screws.Reprinted with permission from Larsson and Bauer [45].

With a recently reported phosphoserine-modified degradable CPC im-proving the adhesion between tissue 40-fold in wet condition compared tocommercial cyanoacrylates glue [60], the contribution of adhesive bondingbetween the bone-CPC and screw-CPC interfaces to the initial engagementof screw implant might lead to a promising augmentation material for screwimplants. However, the improvement of adhesion to the screw engagementneeds to be confirmed and the failure mechanism demands further study.

2.4 Imaging and mechanical testingThe combination of imaging with mechanical testing has been widely ap-plied to capture the trabecular bone behavior and to assist mechanical anal-yses of trabecular bone at different levels and at various resolutions (Table3). For instance, the location of trabecular fracture has been found to behighly correlated to the bone whitening under natural light in tests with both

19

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single trabeculae and trabecular structures [61, 62]. Later, imaging tech-niques such as micro-CT have captured the internal deformation in the tra-becular bone during the mechanical testing, providing more information thanonly the surface changes as recorded by cameras or microscopes. Spatialdevelopment of the internal cracks induced during the trabecular compres-sion is thus possible to reveal [63].

Further digital image processing of the acquired images enables quantita-tive analyses of the trabecular bone. Features such as cracks can be segment-ed and calculated in volume [63]. Techniques such as point tracking or digi-tal image correlation have been adopted to estimate the displacement anddeformation on the trabecular surface without the adverse contribution of theend artifacts [26]. An extension of the image correlation to 3D volume corre-lation allows quantification of volumetric deformations in the trabecularbone.

The accuracy of the image processing methods depends on both the imagequality and the accuracy of the algorithms. The low accessibility to high-resolution images during mechanical loading and digital volume correlation(DVC) algorithms of relatively low accuracy have limited the quantitativeanalyses of the mechanics in trabecular bone undergoing physical defor-mation. Recently, images of high quality and resolution by synchrotron radi-ation based X-ray micro-CT (SRμCT) have become more readily available.A DVC strategy that estimates the global displacement field based on finiteelements has further shown improved performance to the previous localstrategy (see section 4.4 for more details) [64, 65]. The use of higher-orderfinite elements might better capture the continuous deformation in the tra-becular bone. The combination may therefore contribute to the quantificationof deformation with higher spatial resolution, for instance, at the single tra-becular level. Further analyses, such as comparison between strains fromfinite element modeling and DVC can therefore be possible.

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Table 3. Imaging techniques that have been applied during mechanical testing formechanical analyses of trabecular bone.

Single trabeculae Trabecular structureHigh-speedcamera

Bone whitening during frac-ture [61]

Bone whitening during fracture [62]

Effect of dehydration on me-chanical properties [26]

Microscope Displacement estimation [66] Fracture of bone-cement composite[67]

HR-pQCT - 3D observation of bone failure [68]Micro-CT - Compression failure of bone [69]

Screw insertion process [70]Bone failure during screw pullout[71]

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3. Summary of aims and objectives

The aim of this thesis was to further the understanding of trabecular bonebehavior at the tissue and apparent levels as well as in combination withimplants. Micro-CT was used to image the internal structure and the me-chanical behavior of the investigated trabecular bone and implants duringloading, providing the basis for the five experimental papers in this thesis.The specific objectives of these five papers are presented below, togetherwith that of the first, review paper:

(1) to review the literature on the elastic modulus of trabecular bone at thetissue-level and to analyze the effect of several factors on this property (Pa-per I);

(2) to propose a global DVC code based on higher-order finite elementsto estimate internal displacement fields and strains from volumetric imagesof inhomogeneous materials, such as trabecular bone (Paper II);

(3) to propose a method to estimate the elastic modulus of single trabecu-lae based on a combination of imaging during micromechanical testing,global DVC, and finite element analyses (Paper III);

(4) to quantify the strain fields and cracks at the single trabecular levelduring compression of trabecular structure based on in-situ testing and glob-al DVC (Paper IV);

(5) to investigate the effect of two CPC-based augmentation materialswith and without bonding strength, on the pullout resistance of screw im-plants from trabecular bone, in terms of apparent pullout load and crackingmechanisms by SRμCT (Paper V);

(6) to evaluate the feasibility of 3D-printing trabecular structures using Fused Deposition Modelling (FDM) and PLA/HA composites as synthetic trabecular model material (Paper VI).

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4. Methods

4.1 Mechanical testingCompression testingCompression testing is a common experimental method to evaluate the me-chanical properties of various materials, for instance, elastic modulus E,yield strength v, or ultimate compressive strength u. The experiment isperformed by applying a continuous and increasing load to a regular shapedspecimen. The elastic modulus is calculated by E= / , representing the slopeof the linear region of the stress and strain curve. The ultimate compres-sive strength is calculated as u=F/A, where F is the maximum load untilfailure and A is the cross-sectional area.

Three compression tests were performed in this thesis: (1) continuouscompression of cylindrical bulk materials until failure to evaluate the ulti-mate compressive strength (Paper V, VI) at the material level, following theISO 5833 standard for evaluation of acrylic bone cement [72]; (2) compres-sion of single trabeculae as a step towards the estimation of the tissue-levelelastic modulus (Paper III); (3) compression of trabecular structures to eval-uate cracks in trabecular bone (Paper IV) and apparent-level properties ofsynthetic trabecular models (Paper VI). The two tests in Paper III and IV arefurther discussed in section 4.3, as the loading was applied step-wisely, andmicro-CT was incorporated.

Pullout testingThe engagement performance of screw implants in trabecular bone or bonemodels is usually evaluated by pullout testing, where the implants are pulledoutwards from the bone according to a standard method described in ASTMF543 [73]. Due to limitations in specimen dimensions and in hardware suchas motor speed, modified pullout tests were performed in Paper V and VI.The pullout load F is defined as the peak load obtained during the testing,the pullout stiffness is the slope of the linear region of the load and dis-placement curve, and the fracture energy is the area under the load and dis-placement curve until the pullout load F has been reached.

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4

he specimen illuminated by xreconstructed into crosse Fourier

used for further image processing, visualization or calc

Figure

XTwo types of microsources:microfromphoton fluxalmost parallel, and ofbeam hardening and cone beam artifacts can be avoidacquire high resolution images within only a few minutes.monochromatictr

24

4.2 Micro

he specimen illuminated by xreconstructed into crosse Fourier

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Figure

X-ray sourcesTwo types of microsources:microfromphoton fluxalmost parallel, and ofbeam hardening and cone beam artifacts can be avoidacquire high resolution images within only a few minutes.monochromatictrast and reduction in

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he specimen illuminated by xreconstructed into crosse Fourier

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Figure 8.

ray sourcesTwo types of microsources:micro-CTfrom the tubephoton fluxalmost parallel, and ofbeam hardening and cone beam artifacts can be avoidacquire high resolution images within only a few minutes.monochromatic

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he specimen illuminated by xreconstructed into crosse Fourier

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. Illustration

ray sourcesTwo types of microsources: a tube

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ray sourcesTwo types of micro

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Illustration of the process of micro

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aphy system was devel

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computed tomography

re of an object based on itsaphy system was devel

ent, the resolution of this technique

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the process of micro

CT systems were used in this thesis based on the XCT system

right, also referred to as SRμCTis of cone shape, continuous

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ol for life scie

projection data of the specimen rotation angles, and then r

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olution is not possible for clinical

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also referred to as SRμCTcontinuous

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beam and the high photon fluxof the acquired images

computed tomographythe internal stru

Over decades of developm hnique

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projection data of the specimrotation angles, and then r

CT.

were used in this thesis based on the XFig.

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based Xphoton flux.

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to reconstruct the in

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were used in this thesis based on the X-raybased

ray beamrelatively low in

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, and it is possible toMoreover, the

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n-

Tomography is an imaging technique used to reconstruct the internal struc-ture of an object based on its projections in different directions. Ever sincethe first commercial X-ray computed tomography system was developed byHounsfield, this technique has shown great impact, especially in clinicaldiagnostics. Over decades of development, the resolution of this techniquehas improved to a few microns (micro-CT) and even of 100 nm [74] (nano-computed tomography), and while this resolution is not possible for clinicalapplications, it has become a powerful tool for life sciences and materialscience.

The main steps of X-ray based micro-CT are shown in Fig. 8, where 2Dprojection data of the specimen illuminated by X-ray are collected at variousrotation angles, and then reconstructed into cross-sectional images by specif-ic algorithms, for instance Fourier-Based reconstruction. Finally, the recon-structed images are used for further image processing, visualization or calcu-lation.

Page 25: Mechanical analyses of trabecular bone and its interaction ...uu.diva-portal.org/smash/get/diva2:1342155/FULLTEXT01.pdf · Since one of the key mechanical properties of trabecular

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All the SRμCT imaging in this thesis (Papers II-VI) was performed at theTOMCAT beamline, Swiss Light Source (SLS), Paul Scherrer Institut (PSI),Switzerland. A front view of the sample stage is shown in Fig. 10 and de-tailed information about the beamline can be found in Stampanoni et al. [78].

Page 26: Mechanical analyses of trabecular bone and its interaction ...uu.diva-portal.org/smash/get/diva2:1342155/FULLTEXT01.pdf · Since one of the key mechanical properties of trabecular

26

mounting the meof the micro

within theon the acquired images to quantify the strain field

volume of initiated cracks during the loading

insmaller relaxation during the holding time of the step

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technique has been applied to study theof

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26

the meof the micro

stepwithin theon the acquired images to quantify the strain field

of initiated cracks during the loadingthe advantages of SRμCT

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the current thesis,

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IV) and pullout of screw

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disc

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and pullout of screw

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testing setup on the sample stage inside thepossible to do multiple scans of the specimen

loading analyses such as

formed on the acquired to quantify the strain fieldthe volume of initiated the loading

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compression of trabeculae(Paper IV) of screw

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setup on the sample stage inside theCT, it to do multiple scans of the specimen

andanalyses such as

formed on the acquired im the strain fieldthe volume of initiated the loading

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intervertebral 85]

mechanical testscompression of

(Paper IV) and of screw

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oxels within

ulae is calcul

sphere that fits the

situ mechanical chanical setup on the sample stage inside theCT, it is to do multiple scans of the specimen

such asformed on the acquired images the strain field

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testscompression of single ae

(Paper IV) and screwgmentation (Paper V)

situ setups covered in this thesis: (a) compressionof single trabeculae, (b) compression of trabecular structure and (c) pullout of

tal voxels within

trabecular thickness (Tb.Th), n the trabeulae is calculated b e. the diameter of

sphere that fits the

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, track theFurther such as

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[84,

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(Paper IV) and pullout gmentation mater

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bone volume fr

diameter of the largestsphere that fits the marrow space

ae in a unit length, d

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over tube, a major benefit in

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inThis technique has been applied to

situ testscompression of single III), compression of tr

(Paper IV) and pullout of

situ setups covered in this thesis: (a) compressionof single trabeculae, (b) compression of trabecular structure and (c) pullout of

volumen

ngth, d

chanical testing setup sample stage inside theCT, it is possible to scans of the specimen

to Further as

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tube, a major benefit from

the smaller relaxation during the holding of the stepthe imaging

ows This technique has been applied to study of concrete

situ III), compression of tr

(Paper IV) and pullout of the trabecular boneillustrated as

situ setups covered in this thesis: (a) compressionof single trabeculae, (b) compression of trabecular structure and (c) pullout of

n the trabe

er of the largest

nit length, d

chanical testing setup on stage inside theCT, it is possible to do scans of the specimen

to track Further analyses

formed on the acquired images to quantify the d

imaging , a major benefit from the testing with

the smaller relaxation during the holding the stepimagingtests with large setups.

of concrete

situ compression of trtrabecular bone

as

situ setups covered in this thesis: (a) compressionof single trabeculae, (b) compression of trabecular structure and (c) pullout of

n the trabe

er of the largest

nit length, d

chanical testing setup on the stage inside theCT, it is possible to do of the specimen

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imaging over with

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with large setups.concrete

shown of microstru

situ mechanical (Paper on of tr

ar bone, as

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n the trabe

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nit length, d

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formed on the acquired images to quantify the strain

μCTsitu with

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chanical testing setup on the inside theCT, it is possible to do multiple specimen

then be peto quantify

CTsitu

the smaller relaxation during the holding time of the loading,of the

situ tests setups.fracture of

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situ setups covered in this thesis: (a) compressionof single trabeculae, (b) compression of trabecular structure and (c) pullout of

calculated as the diamet

number of trabeculae in a u

chanical testing setup on the sample e theCT, it is possible to do multiple scans

defobe pe

, quantify

situ testing μCT isloading,

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local thickness at a point i

calculated as the diamet

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chanical testing setup on the sample stage theCT, it is possible to do multiple scans of

can pe, or y

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(Paper III), compressi trfrom the trabecular

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chanical testing setup on the sample stage CT, it is possible to do multiple scans of the

level can then

, or to y

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ormed

situ setups covered in this thesis: (a) compressionof single trabeculae, (b) compression of trabecular structure and (c) pullout of

· bone volume fraction (BV/TV), bone volume over total volume,computed as the number of bone voxels over the total voxels withinthe VOI [79];

· trabecular thickness (Tb.Th), local thickness at a point in the trabec-ulae is calculated by the ‘sphere-fitting’ method, i.e. the diameter ofthe largest sphere that fits the trabecula [80];

· trabecular spacing (Tb.Sp), calculated as the diameter of the largestsphere that fits the marrow space [80];

· trabecular number (Tb.N), number of trabeculae in a unit length, de-fined as 1/(Tb.Th+ Tb.Sp);

4.3 In-situ mechanical testing within SR CTBy mounting a mechanical testing setup on the sample stage inside thechamber of a micro-CT, it is possible to do multiple scans of the specimen ata step-wise loading condition, and to track the micrometer-level deformationwithin the specimen. Further analyses such as DVC can then be performedon the acquired images to quantify the strain field [81], or to quantify thevolume of initiated cracks during the loading [82].

Besides the advantages of SR CT imaging over tube-based CT as men-tioned in section 4.2, a major benefit from the in-situ testing with SR CT isthe smaller relaxation during the holding time of the step-wise loading,which in turn benefits the contrast and clarity of the imaging. The sufficientchamber space in a SR CT system allows for in-situ tests with large setups.This technique has been applied to study the fracture of concrete [83], de-formation of intervertebral disc [84, 85], and shown influence of microstruc-tures.

In the current thesis, three in-situ mechanical tests were performed withinSR CT: compression of single trabeculae (Paper III), compression of tra-becular bone (Paper IV) and pullout of screws from the trabecular bone withor without two augmentation materials (Paper V), illustrated as Fig. 11.

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27

4.4 Digital volume correlationSince the first proposal of DVC by Bay et al. [86], DVC has been used formultiple applications, for instance to estimate the deformation fields in tra-becular bone, to detect microcracking in concrete [82] and to determine theboundary conditions for micromechanical simulations of cast iron [87]. DVCcompares two image stacks containing volumetric information of an objectunder different states, which are usually referred to as reference volume Sand deformed volume s (Fig. 12). The idea of DVC is to track the positionchange, i.e. displacement, of a material point from the reference volume tothe deformed volume , utilizing the unique grayscale distribution in theobject. A robust method to calculate the displacement = is to min-imize the inverse normalized cross-correlation between the two volumes[88], as

min 1 ( )( )

(1)

where is the displacement field and denotes the Euclidean norm.

Figure 12. Reference volume and deformed volume . The position vector refersto a position of an arbitrary point in the body in its initial (reference) volume andthe position vector refers to the position of the same point in the body in its de-formed configuration. The displacement of the point is then given by the differenceof the two positions. Reprinted from Paper III.

To increase the sensitivity of the cross-correlation to local differences, the reference volume is divided into a set of sub-volumes. Minimization of the cross-correlation function (Eq.1) is then performed on the sub-volumes. If the minimization process is performed separately for each sub-volume using independent parameters for displacement interpolation, the algorithm is re-ferred to as a local DVC method (Fig. 13b). In a global algorithm (Fig. 13c), the displacement field is estimated with all sub-volumes connected and in-terpolated with a shared set of parameters. The continuous displacement fields given by the global algorithms enable the direct differentiation to cal-culate the strain fields. While in the local algorithms, the displacement fields are interpolated and smoothed before calculation of strain fields, due to the potential discontinuities in the displacement fields. Studies have reported improved performance by using a global algorithm [64, 65].

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Figurealgorithm. Reprinted from Paper II.

ution of the correlation method might be too low

nt field, which is obtained by a rigid registration process

f detail in the images.rithm to track the detail with reliability. Commo

and artifacts such as rings or streaks add non

placement fields might allev

28

Figurealgorithm. Reprinted from Paper II.

yscale variation. However, if the subution of the correlation method might be too low

gorithms, the DVC is solved with mu

nt field, which is obtained by a rigid registration process

f detail in the images.rithm to track the detail with reliability. Commo

and artifacts such as rings or streaks add non

placement fields might allev

Figure 13algorithm. Reprinted from Paper II.

yscale variation. However, if the subution of the correlation method might be too low

gorithms, the DVC is solved with mu

nt field, which is obtained by a rigid registration process

f detail in the images.rithm to track the detail with reliability. Commo

and artifacts such as rings or streaks add non

placement fields might allev

13. Illustration showing the difference between a local and a global DVCalgorithm. Reprinted from Paper II.

yscale variation. However, if the subution of the correlation method might be too low

gorithms, the DVC is solved with mu

nt field, which is obtained by a rigid registration process

f detail in the images.rithm to track the detail with reliability. Commo

and artifacts such as rings or streaks add non

placement fields might allev

. Illustration showing the difference between a local and a global DVCalgorithm. Reprinted from Paper II.

correlation

yscale variation. However, if the subution of the correlation method might be too low

gorithms, the DVC is solved with mu

correlation

nt field, which is obtained by a rigid registration processs

f detail in the images.rithm to track the detail with reliability. Commo

and artifacts such as rings or streaks add non

placement fields might allev

. Illustration showing the difference between a local and a global DVCalgorithm. Reprinted from Paper II.

yscale variation. However, if the subution of the correlation method might be too low

gorithms, the DVC is solved with mu

ained by a rigid registration processs

f detail in the images.rithm to track the detail with reliability. Commo

and artifacts such as rings or streaks add non

placement fields might allev

. Illustration showing the difference between a local and a global DVCalgorithm. Reprinted from Paper II.

yscale variation. However, if the subon method might be too low

gorithms, the DVC is solved with mu

ained by a rigid registration processs

f detail in the images.rithm to track the detail with reliability. Commo

and artifacts such as rings or streaks add non

placement fields might allev

. Illustration showing the difference between a local and a global DVCalgorithm. Reprinted from Paper II.

, if the subon method might be too low

gorithms, the DVC is solved with mu

valued displaceme ained by a rigid registration processs

he algorithms but also on the

lity. Commo

ings or streaks add non

o a non

might allev

. Illustration showing the difference between a local and a global DVCalgorithm. Reprinted from Paper II.

due to the lack of gra , if the subtoo large, spatial resol on method might be too low

gorithms, the DVC is solved with mu

ng point is provided by an integervalued displaceme ained by a rigid registration process

he algorithms but also on the

it is difficult for the algo lity. Commo

ings or streaks add non

polation of grayo a non

of smoothing in the dis might allev

. Illustration showing the difference between a local and a global DVCalgorithm. Reprinted from Paper II.

due to the lack of gra , if the subtoo large, spatial resol on method might be too low

gorithms, the DVC is solved with mu

ng point is provided by an integervalued displaceme ained by a rigid registration process

volume using Fahe algorithms but also on the

image quality or amount oit is difficult for the algo lity. Commo

analyses. System noises ings or streaks add non

polation of grayo a non

of smoothing in the dis might allev

. Illustration showing the difference between a local and a global DVC

n of the dis

too large, spatial resol ethod might be too lowher analysis. Ther thms, the DVC is solved with mu

tiple iterations and

lem, inimum rather than global min

ing point is provided by an integerregistration process

n the algorithms but also on the

it is difficult for the algori detail with reliability. Commo

analyses. System noises an or streaks add nonphysical detail to the ima y

he interpolation of grayd to a non

of smoothing in the displacem allev

ithout physical

. Illustration showing the difference between a local and a global DVC

ompromise between crossn of the dis

too large, spatial r might be too lowhms, the DVC is solved with mu

the risk of reac m rather than global mind by an integer

valued displacement field, which i gistration process

n the algorithms but also on theimage quality or am es.

to a non

of smoothing in the displace ight allev

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. Illustration showing the difference between a local and a global DVC

ise between crossn of the dis

too large, spatial resolution of the cor d might be too lowher analysis. Therefore, in som

starting point

tthe risk of reaching l global min

, the star ovided by an integervalued displacement field, whic a rigid registration process

e algorithms but also on theimage quality or amount

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voxel translations, rotati ns.of smoothing in the displace

tested with two

. Illustration showing the difference between a local and a global DVC

etween crossdis

t not be

too large, spatial resolution of the co d might be too low

olution of the

the risk of reaching l n global minnteger

ut also on theimage quality or amount of detail in th

liability. Commo

voxel translatio ations.of smoothing in the displaceme ev

h two

without physical

. Illustration showing the difference between a local and a global DVC

disot be

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of smoothing in the displacement fields

. Illustration showing the difference between a local and a global DVC

ot bedue to the lack of grayscale variation. However

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on of the

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depends not only on t so on the

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volume is subjected t on, for

of smoothing in the displacement fields

. Illustration showing the difference between a local and a global DVC

resolution of the ot be

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her analysis. Therefore, in some al

on of the

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depends not only on t so on the

it is difficult for the algo

analyses. System noises and artifacts such as r

source of error is the interon, for

. Illustration showing the difference between a local and a global DVC

ot bedue to the lack of grayscale variation. However

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on of the

, the startivalued displacement field, which is obt

depends not only on t so on the

it is difficult for the algo

analyses. System noises and artifacts such as r

source of error is the interon, for

voxel translations, rotations or deformations.

. Illustration showing the difference between a local and a global DVC

too large, spatial resolution of the correlatiher analysis. Therefore, in some al

t

, the starting point is providevalued displacement field, which is obt

depends not only on t

it is difficult for the algorithm to track the detail with reliabi

analyses. System noises and artifacts such as rgenerate a

source of error is the interon, for

. Illustration showing the difference between a local and a global DVC

correlation might n

too large, spatial resolution of the correlation method might be ther analysis. Therefore, in some al

. an

, the starting point is providevalued displacement field, which is obtained by a rigid registr

depends not only on t

it is difficult for the algorithm to track the detail with reliabi

analyses. System noises and artifacts such as r

source of error is the inter

. Illustration showing the difference between a local and a global DVC

too large, spatial resolution of the correlation method might be ther analysis. Therefore, in some al

he spatial resoluti

, the starting point is providevalued displacement field, which is obtained by a rigid registr

depends not only on the algorithms but al

it is difficult for the algorithm to track the detail with reliabi

analyses. System noises and artifacts such as r

integer translati

. Illustration showing the difference between a local and a global DVC

placement field.

her analysis. Therefore, in some al

he spatial resoluti

, the starting point is providevalued displacement field, which is obtained by a rigid registr

depends not only on the algorithms but al

analyses. System noises and artifacts such as r

integer translati

However, the choice of sub-volume size is a compromise between cross-correlation performance and spatial resolution of the displacement field. Ifthe sub-volume is too small, a reliable cross-correlation might not be possi-ble due to the lack of grayscale variation. However, if the sub-volume size istoo large, spatial resolution of the correlation method might be too low forfurther analysis. Therefore, in some algorithms, the DVC is solved with mul-tiple iterations and progressively reduced sub-volume sizes [89]. The cross-correlation is started with a relatively large sub-volume size, and the esti-mated displacement field from the current iteration is used as a starting pointfor the next iteration with a smaller sub-volume size. The spatial resolutionof the displacement field is therefore incrementally and stably improved.

In the minimization problem, a good starting point decreases the compu-tation time and the risk of reaching local minimum rather than global mini-mum. In some studies [90], the starting point is provided by an integer-valued displacement field, which is obtained by a rigid registration processfor each sub-volume using Fast-Fourier Transform (FFT).

The DVC performance depends not only on the algorithms but also on theimage quality or amount of detail in the images. For regions with little detail,it is difficult for the algorithm to track the detail with reliability. Commonly,results of the sub-volumes with too low correlation are excluded from furtheranalyses. System noises and artifacts such as rings or streaks add non-physical detail to the images, which may therefore generate a non-physicaldisplacement field. Another source of error is the interpolation of grayscalevalues whenever a sub-volume is subjected to a non-integer translation, forinstance, sub-voxel translations, rotations or deformations. A proper extentof smoothing in the displacement fields might alleviate the non-physicalstrains without severely affecting the physical deformations.

The quality of the results of DVC algorithms are usually tested with twovolumetric datasets, obtained in several ways:

· Repeated scans (double scan) of one specimen without physical trans-lation;

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· Repeated scans (double scan) of one specimen with physical transla-tion;

In the first three cases, the theoretical strain should be zero for the wholevolume; in the last two cases, the theoretical displacement and strain areknown. Therefore, the differences in displacement or strains between theDVC calculated values and the theoretical values are evaluated and deter-mined as accuracy and precision, which refer to the mean and standard devi-ation of the differences, respectively.

In this thesis, a global DVC algorithm based on higher-order finite ele-ments is proposed (Paper II) and then applied to estimate the deformation insingle trabeculae (Paper III) and trabecular bone (Paper IV) based on thehigh-resolution images from SRμCT (Figure 14).

Figure 14. Example of cross-sectional images by SRμCT from single trabecula (left,reprinted from Paper III) and trabecular structure (right, reprinted from Paper IV).

1 mm

· Generating one volumetric dataset by virtually translating the datasetwith known displacement field, usually non-integer/sub-voxel transla-tion;

· Generating one volumetric dataset by virtually deforming the datasetwith known strain field.

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5. Results and discussion

The results of this thesis, as well as specific discussion, are presented as briefsummaries of each paper. The imaging, mechanical and analysis techniquesinvolved in the papers are listed in Table 4.

Table 4

Paper I II III IV V VI× ×

× × × × ×

× × × ×

× ×

× × ×

Micro-CT SRµCTIn-situ mechanical testingEx-situ mechanical testingDVCFinite element analysis ×

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Paper I - Young’s modulus of the trabecularbone at the tissue level: A review

The elastic modulus of trabecular bone reported in the literature lies in awide range, covering two orders of magnitude. With an aim to narrow thisrange, and/or provide possible improvement suggestions for the test meth-ods, trabecular elastic moduli estimated by four different test methods weresummarized, namely single trabeculae tests, ultrasonic tests, nanoindentationand numerical analyses, advantages and limitations of the methods werediscussed.

Micro-mechanical tests such as tension or bending directly on the singletrabeculae are straight forward methods to estimate the mechanical proper-ties of tissue-level properties of trabecular bone. Due to the small size of thesingle trabeculae, general limitations are present in the sample preparationand handling of the trabeculae, which also have an inhomogeneous geome-try, and the very measurement of the trabecular deformation. For tensiletests, specifically, glue attaching the trabeculae to the grips enhances thedifficulties in accurately measuring only the deformation of trabeculae. Fur-thermore, misalignment of the central line of the specimen with the loadingaxis may lead to uncontrolled shear stress. Bending might be less affected bythese factors, however, the elastic modulus calculated from bending is moresensitive to the specimen geometry.

Based on a relationship between the elastic modulus and the velocity ofwave propagation in the specimen, ultrasonic tests are also applied to esti-mate trabecular elastic modulus. The larger dimensions of specimens de-crease the sensitivity to sample handling. However, the ultrasonic waves canbe strongly attenuated due to the high porosity of the trabecular bone, whichcomplicate the estimation of elastic modulus from this test.

Nanoindentation measures the elastic modulus of trabecular bone at asubmicron structural scale, and has been applied across species, anatomicsites, health states of the donors and specimen conditions. The measuredelastic modulus was reported to be affected by factors such as roughness ofthe sample surface and indentation parameters. Another feature that shouldbe pointed out is the indentation locations. Indentations, for instance, nearlocal defects which are of similar scale as the indenter such as lacunae andosteocyte, are excluded for uncontrolled contact area. This might be a reasonfor the generally higher elastic modulus reported by nanoindentation than for

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tests on single trabeculae which contains all the defects. The level of interesttherefore should be considered while presenting/citing the trabecular proper-ties.

Combining mechanical testing on trabecular bone with finite elementsimulation that takes the trabecular geometry into account has been appliedto estimate the tissue-level elastic modulus [91]. The advantages of thismethod are the convenience in sample handling and the relative insensitivityto trabecular geometry. However, limitations present due to assumptions inthe numerical models, e.g. a homogeneous material distribution and a linearelastic mechanical behavior of the trabecular bone.

The elastic modulus reported by the reviewed literature, and conditions ofthe specimens tested were summarized into tables in Paper I, with the report-ed values ranging from 1 to 22.3 GPa, as a result of physiological and patho-logical conditions as well as scale and other methodology related limitations.Distribution of the elastic modulus was also plotted as a function of testingmethod, anatomic sites and species. However, firm conclusions with statisti-cal significance were not able to be drawn. After applying two exclusioncriteria: 1) number of specimens less than six; 2) other limitations that areconsidered to affect the reliability of the test adversely, studies reportingmore reliable data were plotted in Fig. 15, still ranging from 1.2 to 22.3 GPa.It is expected that, with continuous improvements in the resolution and accu-racy of the test methods, and more methods and studies reported, the me-chanical properties of trabecular bone and its variations among testing condi-tions, species and other factors will be better understood.

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Figurethe exclusion criteriastudy remained for each species, and were hence not included for clarity.from Paper

Figurethe exclusion criteriastudy remained for each species, and were hence not included for clarity.from Paper

Figure 15the exclusion criteriastudy remained for each species, and were hence not included for clarity.from Paper

15. Trabecular elastic modulus reported by studies remaining afterthe exclusion criteriastudy remained for each species, and were hence not included for clarity.from Paper I

Trabecular elastic modulus reported by studies remaining afterthe exclusion criteriastudy remained for each species, and were hence not included for clarity.

I [33]

Trabecular elastic modulus reported by studies remaining afterthe exclusion criteriastudy remained for each species, and were hence not included for clarity.

[33].

Trabecular elastic modulus reported by studies remaining afterthe exclusion criteria. For the other species, i.e. dog, rabbit and ovine, only onestudy remained for each species, and were hence not included for clarity.

Trabecular elastic modulus reported by studies remaining afterFor the other species, i.e. dog, rabbit and ovine, only one

study remained for each species, and were hence not included for clarity.

Trabecular elastic modulus reported by studies remaining afterFor the other species, i.e. dog, rabbit and ovine, only one

study remained for each species, and were hence not included for clarity.

Trabecular elastic modulus reported by studies remaining afterFor the other species, i.e. dog, rabbit and ovine, only one

study remained for each species, and were hence not included for clarity.

Trabecular elastic modulus reported by studies remaining afterFor the other species, i.e. dog, rabbit and ovine, only one

study remained for each species, and were hence not included for clarity.

Trabecular elastic modulus reported by studies remaining afterFor the other species, i.e. dog, rabbit and ovine, only one

study remained for each species, and were hence not included for clarity.

Trabecular elastic modulus reported by studies remaining afterFor the other species, i.e. dog, rabbit and ovine, only one

study remained for each species, and were hence not included for clarity.

Trabecular elastic modulus reported by studies remaining afterFor the other species, i.e. dog, rabbit and ovine, only one

study remained for each species, and were hence not included for clarity.

Trabecular elastic modulus reported by studies remaining afterFor the other species, i.e. dog, rabbit and ovine, only one

study remained for each species, and were hence not included for clarity.

Trabecular elastic modulus reported by studies remaining afterFor the other species, i.e. dog, rabbit and ovine, only one

study remained for each species, and were hence not included for clarity.

Trabecular elastic modulus reported by studies remaining afterFor the other species, i.e. dog, rabbit and ovine, only one

study remained for each species, and were hence not included for clarity.

Trabecular elastic modulus reported by studies remaining afterFor the other species, i.e. dog, rabbit and ovine, only one

study remained for each species, and were hence not included for clarity.

Trabecular elastic modulus reported by studies remaining afterFor the other species, i.e. dog, rabbit and ovine, only one

study remained for each species, and were hence not included for clarity.

Trabecular elastic modulus reported by studies remaining after applyingFor the other species, i.e. dog, rabbit and ovine, only one

Reprinted

33

applyingFor the other species, i.e. dog, rabbit and ovine, only one

Reprinted

33

applyingFor the other species, i.e. dog, rabbit and ovine, only one

Reprinted

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Paper II - A global digital volume correlationalgorithm based on higher-order finiteelements: Implementation and evaluation

To estimate internal deformation of trabecular bone, we proposed a globalDVC technique with higher-order finite element (27-node brick elements) tointerpolate the global displacement field, hereafter, referred to as GDVC-UU. Overlapping sub-volumes were added for more robust solutions andcurvature penalization was applied for regularization. The cross-correlationproblem became a minimization with the objective function

min [ ( )] + ( ) , (2)where and are the displacement vector for a sub-volume e and all sub-volumes, respectively, represented by the displacements at the nodes with alldegrees of freedom; ( ) is a normalized cross-correlation value of sub-volume evaluated analogously as

( )( )

( ) = (3) is the total number of sub-volumes, p is the regularization factor to adjust

the weight of curvature penalty, and the curvature penalty ( ) is definedas

( ) = , , d , (4)

where = d is the volume of sub-volume , is the displacementwhich can be represented by , and ( )/ is the spatial derivative.

The algorithm was applied to trabecular datasets undergoing various dis-placement fields in order to evaluate the code’s performance, including

· virtually (Gaussian) deformed trabecular bone,· virtually translated (in sub-voxel) datasets for interpolation error,· double scan for systematic and random errors,· physically deformed datasets for overall performance.Two trabecular datasets with relatively low and high resolution were test-

ed, referred to as Tozzi-2017 and UU-2018 (Fig. 16), respectively. The re-sults were compared with a commercial global DVC module in Avizo(Thermo Fisher Scientific Inc., Massachusetts , U.S.). Here, results from vir-tually deformed trabeculae and double scan are presented to discuss somebasic aspects of the proposed DVC algorithm.

,

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Figureof TozziReprinted from Paper II

werewerethe same node spacingvolume size, while in Avizo it is the same as the subshowed that tbetter than Avizowith the 27

sented insubaccuracy, whilecurplacement field was also penalizedlocally high curvature (

Figureof TozziReprinted from Paper II

Displacements along two lines across the center of the spatial regionwerewerethe same node spacingvolume size, while in Avizo it is the same as the subshowed that tbetter than Avizowith the 27

Effects of the parameterssented insub-accuracy, whilecurvature in the displacement fieldplacement field was also penalizedlocally high curvature (

Figure 16of Tozzi-Reprinted from Paper II

Displacements along two lines across the center of the spatial regionwere deformedwere compared with the exact solutionthe same node spacingvolume size, while in Avizo it is the same as the subshowed that tbetter than Avizowith the 27

Effects of the parameterssented in

-volume sizeaccuracy, while

vature in the displacement fieldplacement field was also penalizedlocally high curvature (

16. Cross sections of the-2007 (144×144×144) and (right) a slice of UU

Reprinted from Paper II

Displacements along two lines across the center of the spatial regioneformed

compared with the exact solutionthe same node spacingvolume size, while in Avizo it is the same as the subshowed that tbetter than Avizowith the 27-

Effects of the parameterssented in Fig

volume sizeaccuracy, while

vature in the displacement fieldplacement field was also penalizedlocally high curvature (

Cross sections of the2007 (144×144×144) and (right) a slice of UU

Reprinted from Paper II

Displacements along two lines across the center of the spatial regioneformed

compared with the exact solutionthe same node spacingvolume size, while in Avizo it is the same as the subshowed that the proposedbetter than Avizo

-node element over the linear base functions used in Avizo.Effects of the parameters

Fig. 17volume size

accuracy, whilevature in the displacement field

placement field was also penalizedlocally high curvature (

Cross sections of the2007 (144×144×144) and (right) a slice of UU

Reprinted from Paper II

Displacements along two lines across the center of the spatial regioneformed with Gaussian displacements

compared with the exact solutionthe same node spacingvolume size, while in Avizo it is the same as the sub

he proposedbetter than Avizo,

node element over the linear base functions used in Avizo.Effects of the parameters

17. For the volumes without any noise or artifacts, tvolume size captured

accuracy, while large subvature in the displacement field

placement field was also penalizedlocally high curvature (

Cross sections of the2007 (144×144×144) and (right) a slice of UU

Reprinted from Paper II [92]

Displacements along two lines across the center of the spatial regionwith Gaussian displacements

compared with the exact solutionthe same node spacingvolume size, while in Avizo it is the same as the sub

he proposedshowing an advantage of the triquadratic base functions

node element over the linear base functions used in Avizo.Effects of the parameters

For the volumes without any noise or artifacts, tcaptured

large subvature in the displacement field

placement field was also penalizedlocally high curvature (

Cross sections of the2007 (144×144×144) and (right) a slice of UU

[92].

Displacements along two lines across the center of the spatial regionwith Gaussian displacements

compared with the exact solutionthe same node spacing. In GDVCvolume size, while in Avizo it is the same as the sub

he proposedshowing an advantage of the triquadratic base functions

node element over the linear base functions used in Avizo.Effects of the parameters

For the volumes without any noise or artifacts, tcaptured

large subvature in the displacement field

placement field was also penalizedlocally high curvature (Fig.

Cross sections of the trabecular datasets2007 (144×144×144) and (right) a slice of UU

.

Displacements along two lines across the center of the spatial regionwith Gaussian displacements

compared with the exact solutionIn GDVC

volume size, while in Avizo it is the same as the subhe proposed algorithm captured the Gaussian displacements

showing an advantage of the triquadratic base functionsnode element over the linear base functions used in Avizo.

Effects of the parameters ofFor the volumes without any noise or artifacts, tcaptured the Gaussian displacement field

large sub-volvature in the displacement field

placement field was also penalized. 17e

trabecular datasets2007 (144×144×144) and (right) a slice of UU

Displacements along two lines across the center of the spatial regionwith Gaussian displacements

compared with the exact solutionIn GDVC

volume size, while in Avizo it is the same as the subalgorithm captured the Gaussian displacements

showing an advantage of the triquadratic base functionsnode element over the linear base functions used in Avizo.

of GDVCFor the volumes without any noise or artifacts, t

the Gaussian displacement fieldvolume size could not

vature in the displacement fieldplacement field was also penalized

e-f).

trabecular datasets2007 (144×144×144) and (right) a slice of UU

Displacements along two lines across the center of the spatial regionwith Gaussian displacements

compared with the exact solutionIn GDVC-UU,

volume size, while in Avizo it is the same as the subalgorithm captured the Gaussian displacements

showing an advantage of the triquadratic base functionsnode element over the linear base functions used in Avizo.

GDVCFor the volumes without any noise or artifacts, t

the Gaussian displacement fieldume size could not

vature in the displacement fields (placement field was also penalized using

f).

trabecular datasets2007 (144×144×144) and (right) a slice of UU

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compared with the exact solutions andUU,

volume size, while in Avizo it is the same as the subalgorithm captured the Gaussian displacements

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(Figusing

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and the displacements fromUU, the node spacing is half the sub

volume size, while in Avizo it is the same as the subalgorithm captured the Gaussian displacements

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UU onFor the volumes without any noise or artifacts, t

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Fig. 17using regularization, especially in the

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the displacements fromhe node spacing is half the sub

volume size, while in Avizo it is the same as the subalgorithm captured the Gaussian displacements

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17c-d)regularization, especially in the

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the displacements fromhe node spacing is half the sub

volume size, while in Avizo it is the same as the sub-algorithm captured the Gaussian displacements

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the performance are also prFor the volumes without any noise or artifacts, t

the Gaussian displacement fieldume size could not capture

d). The curvature of the diregularization, especially in the

used in this work2007 (144×144×144) and (right) a slice of UU-2018 (200×200×200).

Displacements along two lines across the center of the spatial regionwere evaluated

the displacements fromhe node spacing is half the sub

-volume size.algorithm captured the Gaussian displacements

showing an advantage of the triquadratic base functionsnode element over the linear base functions used in Avizo.

the performance are also prFor the volumes without any noise or artifacts, t

the Gaussian displacement fieldcapture

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used in this work2018 (200×200×200).

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volume size.algorithm captured the Gaussian displacements

showing an advantage of the triquadratic base functionsnode element over the linear base functions used in Avizo.

the performance are also prFor the volumes without any noise or artifacts, t

the Gaussian displacement fieldcapture

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used in this work2018 (200×200×200).

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the displacements fromhe node spacing is half the sub

volume size.algorithm captured the Gaussian displacements

showing an advantage of the triquadratic base functionsnode element over the linear base functions used in Avizo.

the performance are also prFor the volumes without any noise or artifacts, t

the Gaussian displacement fieldsthe locally high

The curvature of the diregularization, especially in the

used in this work: (left) a slice2018 (200×200×200).

Displacements along two lines across the center of the spatial region. The results

the displacements fromhe node spacing is half the sub

volume size. Figalgorithm captured the Gaussian displacements

showing an advantage of the triquadratic base functionsnode element over the linear base functions used in Avizo.

the performance are also prFor the volumes without any noise or artifacts, the smallest

with highestthe locally high

The curvature of the diregularization, especially in the

: (left) a slice2018 (200×200×200).

Displacements along two lines across the center of the spatial regionhe results

the displacements from Avizohe node spacing is half the sub

Fig. 17algorithm captured the Gaussian displacements

showing an advantage of the triquadratic base functionsnode element over the linear base functions used in Avizo.

the performance are also prhe smallest

with highestthe locally high

The curvature of the diregularization, especially in the

35

: (left) a slice2018 (200×200×200).

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Avizo athe node spacing is half the sub

17a-algorithm captured the Gaussian displacements

showing an advantage of the triquadratic base functions

the performance are also pre-he smallest

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35

: (left) a slice2018 (200×200×200).

thathe results

athe node spacing is half the sub-

-balgorithm captured the Gaussian displacements

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s-regularization, especially in the

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36

Figure 17. A cross section of the resulting displacement fields u(x, y) in [voxels]along two lines A and B obtained by Avizo and GDVC-UU for a virtual Gaussiandisplacement field applied to the Tozzi-2017 dataset. Figures (a) and (b) comparethe different methods, (c) and (d) different sub-volume sizes and (e) and (f) differentamounts of regularization. The ’sv’ denotes the sub-volume size. Reprinted fromPaper II [92].

The two algorithms performed comparably at the same node spacing onimages from two repeated scans. It is also noted that accuracy and precisiondecreased with decreasing sub-volume size (Fig. 18), which is contrary tothe Gaussian cases. The increased freedom in the displacement field usingsmaller sub-volumes allows increased fitting to the physical local defor-

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37

mation, but also increased fitting to the noise or artifacts, leading to non-physical deformation.

Regularization is needed to alleviate the non-physical deformation, andthe amount of regularization depends on the pattern and quality of the imag-es. To limit the non-physical strains to below 1‰ in the Tozzi-2017 doublescan, a regularization factor larger than 10 was needed, for the UU-2018dataset, a factor of 104 was needed due to the relatively larger void in onesub-volume.

Figure 18. The accuracy and precision of the calculated strain fields in [‰] ob-tained by Avizo and GDVC-UU for the Tozzi-2017 double scan (152×152×432) andthe UU-2018 double scan. Reprinted from Paper II [92].

Results showed that the proposed global DVC algorithm performs compa-rably to and slightly better than Avizo in most cases. Besides, the resultsshowed that to evaluate the performance of a DVC code in terms of sensi-tivity to noise and artifacts, interpolation error, etc., several tests cases areneeded, including double scan, virtually deformed and physically deformeddatasets.

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38

Paper III - A combined experimental andnumerical method to estimate the elasticmodulus of single trabeculae

The previous studies in Papers I and II demonstrated i) a need to estimate thetissue-level elastic modulus of trabecular bone with higher accuracy, and ii)the possibility of estimating internal displacement of tested specimens withhigh accuracy by a global DVC method. A strategy combing compression ofsingle trabeculae, DVC and finite element (FE) analyses was therefore pro-posed to estimate the elastic modulus of single trabeculae. As illustrated byFig. 19, the strategy started from an in-situ compression test of single tra-beculae, in which the reaction forces of the trabeculae were recorded, andthe trabecular deformation was captured by SRμCT during step-wise load-ing. DVC on the SRμCT images numerically estimated the displacementfields, which then served as boundary conditions in FE models. The high-resolution FE models generated from the SRμCT images accounted for theirregular geometries of the trabeculae. The elastic moduli were finally esti-mated by comparing the reaction forces from the compression and the FEmodels at similar deformations.

Here, two main aspects of this method are discussed and the results onthree trabeculae are presented.

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Figuresingle trabeculae.

ciaas discussed earlier, highly affect the performance.strategy based on virtual strain energy wasphysical

whereand ν=0.3, for simplicity.varied with the choice of parameters. The curvaturereflectfieldcrease in the curvaturereached. The regularization factor for this subat the value ofsubcy and precision in the subsenmaterial

Figuresingle trabeculae.

The first point is the choice of parameterscially subas discussed earlier, highly affect the performance.strategy based on virtual strain energy wasphysical

whereand ν=0.3, for simplicity.varied with the choice of parameters. The curvaturereflectfieldcrease in the curvaturereached. The regularization factor for this subat the value ofsub-cy and precision in the subsen.material

Figure 19single trabeculae.

The first point is the choice of parametersly sub

as discussed earlier, highly affect the performance.strategy based on virtual strain energy wasphysical

=whereand ν=0.3, for simplicity.varied with the choice of parameters. The curvaturereflectsfield concrease in the curvaturereached. The regularization factor for this subat the value of

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. By doing this,material

19. Schematic: Estimating the elastic modulus from compression testing ofsingle trabeculae.

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as discussed earlier, highly affect the performance.strategy based on virtual strain energy wasphysical deformation, let the virtual strain energy

[=

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the stability ofconsiderably

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the stability ofsiderably

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the stability of (, leading to

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Schematic: Estimating the elastic modulus from compression testing offrom Paper III.

The first point is the choice of parametersvolume size and regularization factor

as discussed earlier, highly affect the performance.strategy based on virtual strain energy was

deformation, let the virtual strain energy

, Ω is the volume, repeated indand ν=0.3, for simplicity. The strain energyvaried with the choice of parameters. The curvature

)., leading to

indicatesreached. The regularization factor for this sub

resulting in this firsvolume size and regularization factor, the

cy and precision in the sub-voxel translation test (see sectiona balanced combination

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, Ω is the volume, repeated indThe strain energy

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, leading to lowindicates

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as discussed earlier, highly affect the performance.strategy based on virtual strain energy was

deformation, let the virtual strain energy

, Ω is the volume, repeated indThe strain energy

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indicates areached. The regularization factor for this sub

resulting in this first peakvolume size and regularization factor, the

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Schematic: Estimating the elastic modulus from compression testing ofrom Paper III.

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as discussed earlier, highly affect the performance.strategy based on virtual strain energy was

deformation, let the virtual strain energy

, Ω is the volume, repeated indThe strain energy

varied with the choice of parameters. The curvatureGenerally, a high

strain energya plateau in the strain fields has been

reached. The regularization factor for this subt peak

volume size and regularization factor, thevoxel translation test (see section

a balanced combinationa relatively low virtual strain energy

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as discussed earlier, highly affect the performance.strategy based on virtual strain energy was applied. For a volume undergoing

deformation, let the virtual strain energy

, Ω is the volume, repeated indThe strain energy

varied with the choice of parameters. The curvatureGenerally, a high

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reached. The regularization factor for this subt peak. A

volume size and regularization factor, thevoxel translation test (see section

a balanced combination was obtainedvirtual strain energy

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as discussed earlier, highly affect the performance.applied. For a volume undergoing

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, Ω is the volume, repeated indThe strain energy computed from DVC results

varied with the choice of parameters. The curvatureGenerally, a high

strain energyplateau in the strain fields has been

reached. The regularization factor for this sub-volume size was then chosenAmong several combinations of

volume size and regularization factor, the onevoxel translation test (see section

was obtainedvirtual strain energy

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of the DVC algorithm, esp(here denoted as

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, Ω is the volume, repeated indcomputed from DVC results

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volume size was then chosenmong several combinations ofone that g

voxel translation test (see sectionwas obtained

virtual strain energy

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the DVC algorithm, esp(here denoted as

ere, a straightforwardapplied. For a volume undergoing

(5)ex denote summation

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the DVC algorithm, esp(here denoted as

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39

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39

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40

accuracy in the estimated displacement fieldsubselected

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ulae

orm differently

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ded.

40

accuracy in the estimated displacement fieldsub-selected

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selected

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selected for subs

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accuracy in the estimated displacement fieldvolume size of 40×40×40 voxels and a regularization factor of 100 was

for subs

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regularization factor(bottom)

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for subsequent analyses

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d elastic modulus on theulae

enclosed with a type of epoxyglue, which might def

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enclosed with a type of epoxy

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between the FE analyses and the DVC are p

The more scattered dependency of the estimated e

combination ofvolume size of 40×40×40 voxels and a regularization factor of 100 was

sizes andhe mean and

applying various com-voxel transla-

. The bold lines indicate where the curvature κ has its first peak

The other point is the dependency of the estimate e

the trabecula was enclosed with a ty

between the FE analyses and the DVC are p

The more scattered dependency of the estimated e

combination ofvolume size of 40×40×40 voxels and a regularization factor of 100 was

sizes andhe mean and

m-a-

. The bold lines indicate where the curvature κ has its first peak

The other point is the dependency of the estimate

the trabecula was enclosed with a tythe trabecula. Therefore,

between the FE analyses and the DVC are p

The other point is the dependency of the estimated elastic modulus on thelength of the discretized trabeculae in the FE model. In the experiment, thetip of the trabecula sometimes deformed severely during compression. Addi-tionally, the other end of the trabecula was enclosed with a type of epoxyglue, which might deform differently from the trabecula. Therefore, bothends of the trabecula were gradually excluded from the FE model, with λdenoting the length ratio between the discretized trabeculae and the physicaltrabecula (Fig. 21). The relative difference in cross-sectional displacementbetween the FE analyses and the DVC are plotted for each cross sectionalong the longitudinal (trabecular length) direction, calculated as

= , (6)

where is the longitudinal displacement from FE analyses, is the longi-tudinal displacement from DVC, the superscript denotes all nodal values incross section in the two displacement fields.

Figure 22b shows that the relative difference δ decreased with decreasingλ, as the trabecular slenderness decreased, and more glue and damaged tipregions were excluded.

The more scattered dependency of the estimated elastic moduli on λ isshown in Fig. 22c. After excluding the results from FE models giving rela-

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tive differenceavetrabecular samples. Thoughproposedof

Figurelength λL.

Figureeach cross section along the trabeculae for λ=0.2;0.4;0.6;0.8. c) estimated elasticmoduli at each λ. The black and white markers indicate which values were selectedfor continued anf

tive differenceaverage elastic modulus estimated watrabecular samples. Thoughproposedof single trabecula

Figurelength λL.

Figureeach cross section along the trabeculae for λ=0.2;0.4;0.6;0.8. c) estimated elasticmoduli at each λ. The black and white markers indicate which values were selectedfor continued anfrom Paper III.

tive differencerage elastic modulus estimated wa

trabecular samples. Thoughproposed

single trabecula

Figure 21length λL.

Figure 22each cross section along the trabeculae for λ=0.2;0.4;0.6;0.8. c) estimated elasticmoduli at each λ. The black and white markers indicate which values were selectedfor continued anrom Paper III.

tive differencerage elastic modulus estimated wa

trabecular samples. Thoughproposed method shows a

single trabecula

21. The physical trabecula of length L, and the discretized trabecula oflength λL. Reprinted

22. a) Trabecular shapes from initial CT scans; b) relative differences δ ofeach cross section along the trabeculae for λ=0.2;0.4;0.6;0.8. c) estimated elasticmoduli at each λ. The black and white markers indicate which values were selectedfor continued anrom Paper III.

tive difference δrage elastic modulus estimated wa

trabecular samples. Thoughmethod shows a

single trabecula

The physical trabecula of length L, and the discretized trabecula ofReprinted

a) Trabecular shapes from initial CT scans; b) relative differences δ ofeach cross section along the trabeculae for λ=0.2;0.4;0.6;0.8. c) estimated elasticmoduli at each λ. The black and white markers indicate which values were selectedfor continued analyses.rom Paper III.

δ largerrage elastic modulus estimated wa

trabecular samples. Thoughmethod shows a

single trabeculae

The physical trabecula of length L, and the discretized trabecula ofReprinted f

a) Trabecular shapes from initial CT scans; b) relative differences δ ofeach cross section along the trabeculae for λ=0.2;0.4;0.6;0.8. c) estimated elasticmoduli at each λ. The black and white markers indicate which values were selected

alyses.

argerrage elastic modulus estimated wa

trabecular samples. Thoughmethod shows a

e.

The physical trabecula of length L, and the discretized trabecula offrom Paper III.

a) Trabecular shapes from initial CT scans; b) relative differences δ ofeach cross section along the trabeculae for λ=0.2;0.4;0.6;0.8. c) estimated elasticmoduli at each λ. The black and white markers indicate which values were selected

alyses. X

arger than 10% for over 20% of the trabecular lengthrage elastic modulus estimated wa

trabecular samples. Thoughmethod shows a

The physical trabecula of length L, and the discretized trabecula ofrom Paper III.

a) Trabecular shapes from initial CT scans; b) relative differences δ ofeach cross section along the trabeculae for λ=0.2;0.4;0.6;0.8. c) estimated elasticmoduli at each λ. The black and white markers indicate which values were selected

X3 is the coordinate in the longitudinal direction.

than 10% for over 20% of the trabecular lengthrage elastic modulus estimated wa

trabecular samples. Though this study wasmethod shows a potential

The physical trabecula of length L, and the discretized trabecula ofrom Paper III.

a) Trabecular shapes from initial CT scans; b) relative differences δ ofeach cross section along the trabeculae for λ=0.2;0.4;0.6;0.8. c) estimated elasticmoduli at each λ. The black and white markers indicate which values were selected

is the coordinate in the longitudinal direction.

than 10% for over 20% of the trabecular lengthrage elastic modulus estimated wa

this study waspotential

The physical trabecula of length L, and the discretized trabecula ofrom Paper III.

a) Trabecular shapes from initial CT scans; b) relative differences δ ofeach cross section along the trabeculae for λ=0.2;0.4;0.6;0.8. c) estimated elasticmoduli at each λ. The black and white markers indicate which values were selected

is the coordinate in the longitudinal direction.

than 10% for over 20% of the trabecular lengthrage elastic modulus estimated wa

this study waspotential for

The physical trabecula of length L, and the discretized trabecula of

a) Trabecular shapes from initial CT scans; b) relative differences δ ofeach cross section along the trabeculae for λ=0.2;0.4;0.6;0.8. c) estimated elasticmoduli at each λ. The black and white markers indicate which values were selected

is the coordinate in the longitudinal direction.

than 10% for over 20% of the trabecular lengthrage elastic modulus estimated was 3.83 ± 0.54 GPa for three human

this study wasfor estimat

The physical trabecula of length L, and the discretized trabecula of

a) Trabecular shapes from initial CT scans; b) relative differences δ ofeach cross section along the trabeculae for λ=0.2;0.4;0.6;0.8. c) estimated elasticmoduli at each λ. The black and white markers indicate which values were selected

is the coordinate in the longitudinal direction.

than 10% for over 20% of the trabecular lengths 3.83 ± 0.54 GPa for three human

this study was limited by the sample size, theestimat

The physical trabecula of length L, and the discretized trabecula of

a) Trabecular shapes from initial CT scans; b) relative differences δ ofeach cross section along the trabeculae for λ=0.2;0.4;0.6;0.8. c) estimated elasticmoduli at each λ. The black and white markers indicate which values were selected

is the coordinate in the longitudinal direction.

than 10% for over 20% of the trabecular lengths 3.83 ± 0.54 GPa for three human

limited by the sample size, theestimating

The physical trabecula of length L, and the discretized trabecula of

a) Trabecular shapes from initial CT scans; b) relative differences δ ofeach cross section along the trabeculae for λ=0.2;0.4;0.6;0.8. c) estimated elasticmoduli at each λ. The black and white markers indicate which values were selected

is the coordinate in the longitudinal direction.

than 10% for over 20% of the trabecular lengths 3.83 ± 0.54 GPa for three human

limited by the sample size, theing the mechanical properties

The physical trabecula of length L, and the discretized trabecula of

a) Trabecular shapes from initial CT scans; b) relative differences δ ofeach cross section along the trabeculae for λ=0.2;0.4;0.6;0.8. c) estimated elasticmoduli at each λ. The black and white markers indicate which values were selected

is the coordinate in the longitudinal direction.

than 10% for over 20% of the trabecular lengths 3.83 ± 0.54 GPa for three human

limited by the sample size, thethe mechanical properties

The physical trabecula of length L, and the discretized trabecula of

a) Trabecular shapes from initial CT scans; b) relative differences δ ofeach cross section along the trabeculae for λ=0.2;0.4;0.6;0.8. c) estimated elasticmoduli at each λ. The black and white markers indicate which values were selected

is the coordinate in the longitudinal direction.

than 10% for over 20% of the trabecular lengths 3.83 ± 0.54 GPa for three human

limited by the sample size, thethe mechanical properties

The physical trabecula of length L, and the discretized trabecula of

a) Trabecular shapes from initial CT scans; b) relative differences δ ofeach cross section along the trabeculae for λ=0.2;0.4;0.6;0.8. c) estimated elasticmoduli at each λ. The black and white markers indicate which values were selected

is the coordinate in the longitudinal direction.

than 10% for over 20% of the trabecular lengths 3.83 ± 0.54 GPa for three human

limited by the sample size, thethe mechanical properties

The physical trabecula of length L, and the discretized trabecula of

a) Trabecular shapes from initial CT scans; b) relative differences δ ofeach cross section along the trabeculae for λ=0.2;0.4;0.6;0.8. c) estimated elasticmoduli at each λ. The black and white markers indicate which values were selected

is the coordinate in the longitudinal direction.

than 10% for over 20% of the trabecular lengths 3.83 ± 0.54 GPa for three human

limited by the sample size, thethe mechanical properties

The physical trabecula of length L, and the discretized trabecula of

a) Trabecular shapes from initial CT scans; b) relative differences δ ofeach cross section along the trabeculae for λ=0.2;0.4;0.6;0.8. c) estimated elasticmoduli at each λ. The black and white markers indicate which values were selected

is the coordinate in the longitudinal direction.

than 10% for over 20% of the trabecular lengths 3.83 ± 0.54 GPa for three human

limited by the sample size, thethe mechanical properties

The physical trabecula of length L, and the discretized trabecula of

a) Trabecular shapes from initial CT scans; b) relative differences δ ofeach cross section along the trabeculae for λ=0.2;0.4;0.6;0.8. c) estimated elasticmoduli at each λ. The black and white markers indicate which values were selected

Reprinted

41

than 10% for over 20% of the trabecular length, thes 3.83 ± 0.54 GPa for three human

limited by the sample size, thethe mechanical properties

The physical trabecula of length L, and the discretized trabecula of

a) Trabecular shapes from initial CT scans; b) relative differences δ ofeach cross section along the trabeculae for λ=0.2;0.4;0.6;0.8. c) estimated elasticmoduli at each λ. The black and white markers indicate which values were selected

Reprinted

41

, thes 3.83 ± 0.54 GPa for three human

limited by the sample size, thethe mechanical properties

The physical trabecula of length L, and the discretized trabecula of

a) Trabecular shapes from initial CT scans; b) relative differences δ ofeach cross section along the trabeculae for λ=0.2;0.4;0.6;0.8. c) estimated elasticmoduli at each λ. The black and white markers indicate which values were selected

Reprinted

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42

Paper IV - Quantification of strains and cracksin trabecular bone by digital volumecorrelation: A case study

The combination of step-wise loading, micro-CT and DVC has enabled thevisualization and quantification of internal deformations of trabecular struc-tures. The local strain fields can provide a ‘cross-validation’ for finite ele-ment models of these structures, which have been widely applied to simulatebone mechanical behavior [93-95]. Whilst deformation, cracks and damageaccumulation have been observed locally on individual trabeculae in the teststructures, strains have seldom been estimated at this level due to relativelylow image resolutions (20-40 µm) and therefore too low DVC spatial resolu-tions (0.8-2 mm) to investigate single trabeculae. Recently, local strains wereestimated by DVC on high-resolution images from SRµCT of trabecularbone loaded in the apparent elastic range [75]. The aim of this case studywas to quantify trabecular strains and cracks during compression until failureusing the proposed DVC algorithm (Paper II) on SRµCT images.

An in-situ compression test was performed on a human trabecular cylin-der, which was scanned at a voxel size of 3.25 µm. The sub-volume size andregularization factor of the DVC method, 64 and 60 respectively, were se-lected based on a test-volume of 200×200×200 voxels using the same strate-gy as discussed in Paper III, i.e. determined through a balance of low virtualstrain energy and high accuracy.

Figure 23 shows a general increase in the first principle strains, and high-er strains near the convex surface where displacement was gradually applied(Fig. 23). Strain concentrations were observed in trabeculae at a distancefrom the top, indicating an effect of the discrete nature of the trabecular ar-chitecture to the strain distribution. When zoomed in to the highly deformedtrabeculae, we noticed that individual trabeculae were subjected to high ten-sile and shear strains though the trabecular bone was under compression atthe apparent level. Predominant open mode fractures and some few shearmode fractures were found by contrasting the cracks and tensile/shearstrains.

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4343

Figure 23. The first principle strain (EI) of the trabecular specimen at loading steps2-5. Displacement was applied in the negative Z direction. Strain was plotted in theinitial configurations. Reproduced from Paper IV.

Cracks in the trabecular bone were segmented with DVC-based image processing. Based on step 6 where an obvious increase in crack volume was noticed (Fig. 24), the trabecular structure was divided into cracked and ‘uncracked’ regions. Mean and standard deviations of the first principal strains in the two groups were contrasted at steps 2-6. Regions in the trabecu-lae that had visible cracks in step 6 showed higher strains than the uncracked regions. The difference between the two groups increased slightly from step 2 to step 5, and was a factor of 7.6 at step 6. When the cracks in step 6 were grouped by the crack volume, larger opening cracks showed higher tensile strains than smaller cracks. Note that not all regions with high strains resulted in cracks and some trabeculae with low strain in step 5 were observed with cracks in step 6.

Figure 24. Load and displacement at the load steps in the uniaxial compressiontesting, as well as crack volumes estimated by a DVC-based image processing andnormalized by the total bone volume (left). Mean and standard deviations of the firstprincipal strain at steps 2-6 in the cracked and ‘uncracked’ regions. Reproducedfrom Paper IV.

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44

Paper V - The effect of two augmentationmaterials on screw pullout resistance fromhuman trabecular bone

Augmentation materials, such as CPCs and PMMA, have been used to im-prove the physical engagement of screws inserted into bone. While ceramic,degradable cements may ultimately improve the fixation [49, 51, 96], incon-sistent effects on early stability have been reported [57].

A recently developed degradable ceramic adhesive has been reported tobe able to bond with tissues with relatively high strength [60]. Consideringthat the bonding of screws with surrounding bone may contribute to moreevenly distributed loads in the vicinity of the screw and along the threads,augmentation with the adhesive may improve the pullout performance. Thepullout resistance may be further enhanced by transiting to adhesive bondingfrom physical contact at the interfaces, with improved shear and tensile frac-ture toughness. The aim of this study was to investigate the screw pulloutresistance and failure mechanisms from trabecular bone, and the effect oftwo augmentation materials, i.e. a ceramic cement or an adhesive material,with no augmentation as a control.

Two pullout tests were performed, one of which was an ex-situ continu-ous pullout of screws from 21 trabecular specimens which were divided intothree groups by the augmentation materials, i.e. cement, adhesive and noaugmentation as a control (Fig. 25). The other one was an in-situ step-wisepullout inside a SRμCT. The five specimens for the in-situ testing werenamed following the form: augmentation material-in-specimen number.

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45

Figure 25. Section views from micro-CT scans showing the inserted screws and thedistribution of the augmentation materials in ex-situ specimens in three groups: (a)control, (b) cement, (c) adhesive. Reprinted from Paper V.

Two parameters were characterized from the micro-CT images: BV/TV and injection volume of the augmentation materials. BV/TV was calculated from the images acquired before the screw insertion. The injection volume was estimated based on the comparison of micro-CT images before and after the screw insertion.

Linear regression showed the highest correlation of the pullout load withBV/TV in the region just around the screw. The high correlation was in ac-cordance with the earlier reported main effect of BV/TV [18, 97] or bonemineral density [98, 99] on the pullout strength, confirming the contributionof trabecular bone in the direct vicinity of the screw to the pullout resistance.

The pullout load, stiffness and fracture energy (calculated as the area un-der the load-displacement curve until maximum load) of the ex-situ testing normalized with BV/TV were compared among the three groups by one-way ANOVA (Fig. 26). Augmentation with the cement did not affect the pullout resistance, compared to the control group. In contrast, the pullout resistance after augmentation with the adhesive was significantly higher than the cement and the control group. This enhancement is comparable to the literature-reported enhancement after augmentation with ceramic cement, but slightly worse than with PMMA in vertebrae (1.8-2.8 fold [49, 56, 100]).

Note that the injection volume and the distribution of the two augmenta-tion material were different (Fig. 25), which might affect the pullout re-sistance. When the pullout properties were normalized with a factor thatincluded both BV/TV and injection volume, there were no significant differ-ences in most cases. However, the linear relation between pullout propertieswith this factor needs to be verified. On the other hand, the adhesive showedan advantage in injectability and material distribution, compared to thebrushite cement used.

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46

Figurefracture energy of the exing for BV/TV. The asterisk mark (*) denotes significant difference between twocompared groups.

ing steps are exemplified for one specimen with/without augmentation (27propagate[101]augmented with the cementadhesivetrabeculaand theperipheries of thethe adhesive and trabeculae near the screw.

FigureControlplacement in mm

46

Figurefracture energy of the exing for BV/TV. The asterisk mark (*) denotes significant difference between twocompared groups.

Slices of the periing steps are exemplified for one specimen with/without augmentation (27-30propagate[101]augmented with the cementadhesivetrabeculaand theperipheries of thethe adhesive and trabeculae near the screw.

FigureControlplacement in mm

Figure 26fracture energy of the exing for BV/TV. The asterisk mark (*) denotes significant difference between twocompared groups.

Slices of the periing steps are exemplified for one specimen with/without augmentation (

30). In a specimen without augmentationpropagate[101]. Laugmented with the cementadhesivetrabeculaand theperipheries of thethe adhesive and trabeculae near the screw.

Figure 27Control-inplacement in mm

26. Estimated marginfracture energy of the exing for BV/TV. The asterisk mark (*) denotes significant difference between twocompared groups.

Slices of the periing steps are exemplified for one specimen with/without augmentation (

). In a specimen without augmentationpropagated

Long cracks across several trabeculae were not observedaugmented with the cementadhesive (Adhesietrabeculae, and cracks were observed to propagate through the trabeculaeand the adhesperipheries of thethe adhesive and trabeculae near the screw.

27. Crack propagation across several trabeculae during screw pullout fromin-1. ‘B’ denotes bone, ‘S’ denotes the screw

placement in mm

Estimated marginfracture energy of the exing for BV/TV. The asterisk mark (*) denotes significant difference between twocompared groups.

Slices of the periing steps are exemplified for one specimen with/without augmentation (

). In a specimen without augmentationacross the trabeculae, similar to the crack paths in brittle foams

ong cracks across several trabeculae were not observedaugmented with the cement

(Adhesie, and cracks were observed to propagate through the trabeculae

adhesive. These cracks initiated at both the thread crests and theperipheries of thethe adhesive and trabeculae near the screw.

Crack propagation across several trabeculae during screw pullout from1. ‘B’ denotes bone, ‘S’ denotes the screw

placement in mm

Estimated marginfracture energy of the exing for BV/TV. The asterisk mark (*) denotes significant difference between twocompared groups. Reprinted

Slices of the periing steps are exemplified for one specimen with/without augmentation (

). In a specimen without augmentationacross the trabeculae, similar to the crack paths in brittle foams

ong cracks across several trabeculae were not observedaugmented with the cement

(Adhesie-, and cracks were observed to propagate through the trabeculae

ive. These cracks initiated at both the thread crests and theperipheries of the adhesive, indicating an improved load distribution thrthe adhesive and trabeculae near the screw.

Crack propagation across several trabeculae during screw pullout from1. ‘B’ denotes bone, ‘S’ denotes the screw

placement in mm. Reprinted

Estimated marginfracture energy of the ex-ing for BV/TV. The asterisk mark (*) denotes significant difference between two

Reprinted

Slices of the peri-implant region from the ining steps are exemplified for one specimen with/without augmentation (

). In a specimen without augmentationacross the trabeculae, similar to the crack paths in brittle foams

ong cracks across several trabeculae were not observedaugmented with the cement

-in-1,, and cracks were observed to propagate through the trabeculae

ive. These cracks initiated at both the thread crests and theadhesive, indicating an improved load distribution thr

the adhesive and trabeculae near the screw.

Crack propagation across several trabeculae during screw pullout from1. ‘B’ denotes bone, ‘S’ denotes the screw

Reprinted

Estimated margin-situ specimens, given by the general linear model accoun

ing for BV/TV. The asterisk mark (*) denotes significant difference between twoReprinted from Paper

implant region from the ining steps are exemplified for one specimen with/without augmentation (

). In a specimen without augmentationacross the trabeculae, similar to the crack paths in brittle foams

ong cracks across several trabeculae were not observedaugmented with the cement

1, Fig, and cracks were observed to propagate through the trabeculae

ive. These cracks initiated at both the thread crests and theadhesive, indicating an improved load distribution thr

the adhesive and trabeculae near the screw.

Crack propagation across several trabeculae during screw pullout from1. ‘B’ denotes bone, ‘S’ denotes the screw

Reprinted from Paper V.

Estimated marginal means of the (a) pullout load, (b) stiffness and (c)situ specimens, given by the general linear model accoun

ing for BV/TV. The asterisk mark (*) denotes significant difference between twofrom Paper

implant region from the ining steps are exemplified for one specimen with/without augmentation (

). In a specimen without augmentationacross the trabeculae, similar to the crack paths in brittle foams

ong cracks across several trabeculae were not observedaugmented with the cement. However, in the specimen augmented with the

Figs., and cracks were observed to propagate through the trabeculae

ive. These cracks initiated at both the thread crests and theadhesive, indicating an improved load distribution thr

the adhesive and trabeculae near the screw.

Crack propagation across several trabeculae during screw pullout from1. ‘B’ denotes bone, ‘S’ denotes the screw

from Paper V.

al means of the (a) pullout load, (b) stiffness and (c)situ specimens, given by the general linear model accoun

ing for BV/TV. The asterisk mark (*) denotes significant difference between twofrom Paper

implant region from the ining steps are exemplified for one specimen with/without augmentation (

). In a specimen without augmentationacross the trabeculae, similar to the crack paths in brittle foams

ong cracks across several trabeculae were not observed. However, in the specimen augmented with the

27-28, and cracks were observed to propagate through the trabeculae

ive. These cracks initiated at both the thread crests and theadhesive, indicating an improved load distribution thr

the adhesive and trabeculae near the screw.

Crack propagation across several trabeculae during screw pullout from1. ‘B’ denotes bone, ‘S’ denotes the screw

from Paper V.

al means of the (a) pullout load, (b) stiffness and (c)situ specimens, given by the general linear model accoun

ing for BV/TV. The asterisk mark (*) denotes significant difference between twofrom Paper V.

implant region from the ining steps are exemplified for one specimen with/without augmentation (

). In a specimen without augmentationacross the trabeculae, similar to the crack paths in brittle foams

ong cracks across several trabeculae were not observed. However, in the specimen augmented with the

28), the, and cracks were observed to propagate through the trabeculae

ive. These cracks initiated at both the thread crests and theadhesive, indicating an improved load distribution thr

the adhesive and trabeculae near the screw.

Crack propagation across several trabeculae during screw pullout from1. ‘B’ denotes bone, ‘S’ denotes the screw

from Paper V.

al means of the (a) pullout load, (b) stiffness and (c)situ specimens, given by the general linear model accoun

ing for BV/TV. The asterisk mark (*) denotes significant difference between twoV.

implant region from the ining steps are exemplified for one specimen with/without augmentation (

). In a specimen without augmentationacross the trabeculae, similar to the crack paths in brittle foams

ong cracks across several trabeculae were not observed. However, in the specimen augmented with the

, the, and cracks were observed to propagate through the trabeculae

ive. These cracks initiated at both the thread crests and theadhesive, indicating an improved load distribution thr

the adhesive and trabeculae near the screw.

Crack propagation across several trabeculae during screw pullout from1. ‘B’ denotes bone, ‘S’ denotes the screw

from Paper V.

al means of the (a) pullout load, (b) stiffness and (c)situ specimens, given by the general linear model accoun

ing for BV/TV. The asterisk mark (*) denotes significant difference between two

implant region from the ining steps are exemplified for one specimen with/without augmentation (

). In a specimen without augmentationacross the trabeculae, similar to the crack paths in brittle foams

ong cracks across several trabeculae were not observed. However, in the specimen augmented with the

, the adhesive filled the space between, and cracks were observed to propagate through the trabeculae

ive. These cracks initiated at both the thread crests and theadhesive, indicating an improved load distribution thr

the adhesive and trabeculae near the screw.

Crack propagation across several trabeculae during screw pullout from1. ‘B’ denotes bone, ‘S’ denotes the screw

al means of the (a) pullout load, (b) stiffness and (c)situ specimens, given by the general linear model accoun

ing for BV/TV. The asterisk mark (*) denotes significant difference between two

implant region from the in-situ testing at different loaing steps are exemplified for one specimen with/without augmentation (

). In a specimen without augmentation (Controlacross the trabeculae, similar to the crack paths in brittle foams

ong cracks across several trabeculae were not observed. However, in the specimen augmented with the

adhesive filled the space between, and cracks were observed to propagate through the trabeculae

ive. These cracks initiated at both the thread crests and theadhesive, indicating an improved load distribution thr

Crack propagation across several trabeculae during screw pullout from1. ‘B’ denotes bone, ‘S’ denotes the screw, and ‘d’ denotes applied di

al means of the (a) pullout load, (b) stiffness and (c)situ specimens, given by the general linear model accoun

ing for BV/TV. The asterisk mark (*) denotes significant difference between two

situ testing at different loaing steps are exemplified for one specimen with/without augmentation (

(Controlacross the trabeculae, similar to the crack paths in brittle foams

ong cracks across several trabeculae were not observed. However, in the specimen augmented with the

adhesive filled the space between, and cracks were observed to propagate through the trabeculae

ive. These cracks initiated at both the thread crests and theadhesive, indicating an improved load distribution thr

Crack propagation across several trabeculae during screw pullout from, and ‘d’ denotes applied di

al means of the (a) pullout load, (b) stiffness and (c)situ specimens, given by the general linear model accoun

ing for BV/TV. The asterisk mark (*) denotes significant difference between two

situ testing at different loaing steps are exemplified for one specimen with/without augmentation (

(Control-inacross the trabeculae, similar to the crack paths in brittle foams

ong cracks across several trabeculae were not observed. However, in the specimen augmented with the

adhesive filled the space between, and cracks were observed to propagate through the trabeculae

ive. These cracks initiated at both the thread crests and theadhesive, indicating an improved load distribution thr

Crack propagation across several trabeculae during screw pullout from, and ‘d’ denotes applied di

al means of the (a) pullout load, (b) stiffness and (c)situ specimens, given by the general linear model accoun

ing for BV/TV. The asterisk mark (*) denotes significant difference between two

situ testing at different loaing steps are exemplified for one specimen with/without augmentation (

in-1,across the trabeculae, similar to the crack paths in brittle foams

ong cracks across several trabeculae were not observed. However, in the specimen augmented with the

adhesive filled the space between, and cracks were observed to propagate through the trabeculae

ive. These cracks initiated at both the thread crests and theadhesive, indicating an improved load distribution thr

Crack propagation across several trabeculae during screw pullout from, and ‘d’ denotes applied di

al means of the (a) pullout load, (b) stiffness and (c)situ specimens, given by the general linear model accoun

ing for BV/TV. The asterisk mark (*) denotes significant difference between two

situ testing at different loaing steps are exemplified for one specimen with/without augmentation (

Fig.across the trabeculae, similar to the crack paths in brittle foams

ong cracks across several trabeculae were not observed. However, in the specimen augmented with the

adhesive filled the space between, and cracks were observed to propagate through the trabeculae

ive. These cracks initiated at both the thread crests and theadhesive, indicating an improved load distribution thr

Crack propagation across several trabeculae during screw pullout from, and ‘d’ denotes applied di

al means of the (a) pullout load, (b) stiffness and (c)situ specimens, given by the general linear model accoun

ing for BV/TV. The asterisk mark (*) denotes significant difference between two

situ testing at different loaing steps are exemplified for one specimen with/without augmentation (

Fig. 27across the trabeculae, similar to the crack paths in brittle foams

in specimens. However, in the specimen augmented with the

adhesive filled the space between, and cracks were observed to propagate through the trabeculae

ive. These cracks initiated at both the thread crests and theadhesive, indicating an improved load distribution thr

Crack propagation across several trabeculae during screw pullout from, and ‘d’ denotes applied di

al means of the (a) pullout load, (b) stiffness and (c)situ specimens, given by the general linear model accoun

ing for BV/TV. The asterisk mark (*) denotes significant difference between two

situ testing at different loaing steps are exemplified for one specimen with/without augmentation (

27), cracksacross the trabeculae, similar to the crack paths in brittle foams

in specimens. However, in the specimen augmented with the

adhesive filled the space between, and cracks were observed to propagate through the trabeculae

ive. These cracks initiated at both the thread crests and theadhesive, indicating an improved load distribution thr

Crack propagation across several trabeculae during screw pullout from, and ‘d’ denotes applied di

al means of the (a) pullout load, (b) stiffness and (c)situ specimens, given by the general linear model accoun

ing for BV/TV. The asterisk mark (*) denotes significant difference between two

situ testing at different load-ing steps are exemplified for one specimen with/without augmentation (Figs.

, cracksacross the trabeculae, similar to the crack paths in brittle foams

in specimens. However, in the specimen augmented with the

adhesive filled the space between, and cracks were observed to propagate through the trabeculae

ive. These cracks initiated at both the thread crests and theadhesive, indicating an improved load distribution through

Crack propagation across several trabeculae during screw pullout from, and ‘d’ denotes applied dis-

al means of the (a) pullout load, (b) stiffness and (c)situ specimens, given by the general linear model account-

ing for BV/TV. The asterisk mark (*) denotes significant difference between two

d-Figs.

, cracksacross the trabeculae, similar to the crack paths in brittle foams

in specimens. However, in the specimen augmented with the

adhesive filled the space between, and cracks were observed to propagate through the trabeculae

ive. These cracks initiated at both the thread crests and theough

Crack propagation across several trabeculae during screw pullout from

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Figurefrom Adhesive‘d’ denotes applied displacement in mm

Figuresiveand propagated towards the screw. ‘A’ denotesdenotes screw V.

Figurefrom Adhesive‘d’ denotes applied displacement in mm

Figuresive-and propagated towards the screw. ‘A’ denotesdenotes screw V.

Figure 29from Adhesive‘d’ denotes applied displacement in mm

Figure 30-in-1, showing crack initiated at the periphery of the bone

and propagated towards the screw. ‘A’ denotesdenotes screw

29. Crack propagation across trabeculae and adhesive during screw pulloutfrom Adhesive‘d’ denotes applied displacement in mm

30. Induced cracks in bone and adhesive during screw pullout from Adh1, showing crack initiated at the periphery of the bone

and propagated towards the screw. ‘A’ denotesdenotes screw

Crack propagation across trabeculae and adhesive during screw pulloutfrom Adhesive-in‘d’ denotes applied displacement in mm

Induced cracks in bone and adhesive during screw pullout from Adh1, showing crack initiated at the periphery of the bone

and propagated towards the screw. ‘A’ denotesdenotes screw, and ‘d’ denot

Crack propagation across trabeculae and adhesive during screw pulloutin-1. ‘A’ deno

‘d’ denotes applied displacement in mm

Induced cracks in bone and adhesive during screw pullout from Adh1, showing crack initiated at the periphery of the bone

and propagated towards the screw. ‘A’ denotes, and ‘d’ denot

Crack propagation across trabeculae and adhesive during screw pullout1. ‘A’ deno

‘d’ denotes applied displacement in mm

Induced cracks in bone and adhesive during screw pullout from Adh1, showing crack initiated at the periphery of the bone

and propagated towards the screw. ‘A’ denotes, and ‘d’ denot

Crack propagation across trabeculae and adhesive during screw pullout1. ‘A’ deno

‘d’ denotes applied displacement in mm

Induced cracks in bone and adhesive during screw pullout from Adh1, showing crack initiated at the periphery of the bone

and propagated towards the screw. ‘A’ denotes, and ‘d’ denot

Crack propagation across trabeculae and adhesive during screw pullout1. ‘A’ denotes adhesive, ‘B’ denotes bone,

‘d’ denotes applied displacement in mm

Induced cracks in bone and adhesive during screw pullout from Adh1, showing crack initiated at the periphery of the bone

and propagated towards the screw. ‘A’ denotes, and ‘d’ denotes applied displacement in mm

Crack propagation across trabeculae and adhesive during screw pullouttes adhesive, ‘B’ denotes bone,

‘d’ denotes applied displacement in mm

Induced cracks in bone and adhesive during screw pullout from Adh1, showing crack initiated at the periphery of the bone

and propagated towards the screw. ‘A’ denoteses applied displacement in mm

Crack propagation across trabeculae and adhesive during screw pullouttes adhesive, ‘B’ denotes bone,

‘d’ denotes applied displacement in mm. Reprinted

Induced cracks in bone and adhesive during screw pullout from Adh1, showing crack initiated at the periphery of the bone

and propagated towards the screw. ‘A’ denoteses applied displacement in mm

Crack propagation across trabeculae and adhesive during screw pullouttes adhesive, ‘B’ denotes bone,

Reprinted

Induced cracks in bone and adhesive during screw pullout from Adh1, showing crack initiated at the periphery of the bone

and propagated towards the screw. ‘A’ denoteses applied displacement in mm

Crack propagation across trabeculae and adhesive during screw pullouttes adhesive, ‘B’ denotes bone,

Reprinted

Induced cracks in bone and adhesive during screw pullout from Adh1, showing crack initiated at the periphery of the bone

and propagated towards the screw. ‘A’ denoteses applied displacement in mm

Crack propagation across trabeculae and adhesive during screw pullouttes adhesive, ‘B’ denotes bone,

Reprinted from Paper V.

Induced cracks in bone and adhesive during screw pullout from Adh1, showing crack initiated at the periphery of the bone

and propagated towards the screw. ‘A’ denotes adhesive, ‘B’ denotes bone,es applied displacement in mm

Crack propagation across trabeculae and adhesive during screw pullouttes adhesive, ‘B’ denotes bone,

from Paper V.

Induced cracks in bone and adhesive during screw pullout from Adh1, showing crack initiated at the periphery of the bone

adhesive, ‘B’ denotes bone,es applied displacement in mm

Crack propagation across trabeculae and adhesive during screw pullouttes adhesive, ‘B’ denotes bone, ‘S’ denotes screw

from Paper V.

Induced cracks in bone and adhesive during screw pullout from Adh1, showing crack initiated at the periphery of the bone

adhesive, ‘B’ denotes bone,es applied displacement in mm.

Crack propagation across trabeculae and adhesive during screw pullout‘S’ denotes screw

from Paper V.

Induced cracks in bone and adhesive during screw pullout from Adh1, showing crack initiated at the periphery of the bone-adhesive composite

adhesive, ‘B’ denotes bone,Reprinted

Crack propagation across trabeculae and adhesive during screw pullout‘S’ denotes screw

Induced cracks in bone and adhesive during screw pullout from Adhadhesive composite

adhesive, ‘B’ denotes bone,Reprinted

Crack propagation across trabeculae and adhesive during screw pullout‘S’ denotes screw

Induced cracks in bone and adhesive during screw pullout from Adhadhesive composite

adhesive, ‘B’ denotes bone,Reprinted from Paper

Crack propagation across trabeculae and adhesive during screw pullout‘S’ denotes screw

Induced cracks in bone and adhesive during screw pullout from Adhadhesive composite

adhesive, ‘B’ denotes bone,from Paper

47

Crack propagation across trabeculae and adhesive during screw pullout‘S’ denotes screw, and

Induced cracks in bone and adhesive during screw pullout from Adhadhesive composite

adhesive, ‘B’ denotes bone, ‘S’from Paper

47

Crack propagation across trabeculae and adhesive during screw pullout, and

Induced cracks in bone and adhesive during screw pullout from Adhe-adhesive composite

‘S’from Paper

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Paper VI - 3D-printed PLA/HA compositestructures as synthetic trabecular bone: afeasibility study using Fused DepositionModelling

Synthetic models mimicking trabecular architectures may be beneficial forunderstanding the mechanical effects of the trabecular structure in the evalu-ation of implant design for example. The flexibility of 3D-printing tech-niques, together with micro-CT, provides a possibility to reproduce a specif-ic trabecular bone. The aim of this study was to evaluate the feasibility ofprinting model trabecular bone using FDM and PLA/HA composites.

Micro-CT images of trabecular bone taken from a human femoral head were used to generate models for printing, and scaled synthetic bone was printed with PLA/HA composite filaments of three different HA concentra-tions, 5-10-15 wt%. Material characterization and compressive testing of the composite bulk material were performed. Morphology and mechanical prop-erties of the printed models were evaluated with micro-CT, compression and screw pullout tests.

Morphometric analyses showed that the printed PLA model could reproduce profiles of the input trabecular model with accuracy, albeit with significantly higher surface-to-volume ratio and lower trabecular thickness than the input model due to under-filling. In addition, incorpora-tion of HA particles reduced the printing accuracy.

Compression tests showed a significantly increased elastic modulus of theprinted bulk material with the incorporation of HA particles compared topure PLA. No significant difference in compressive yield strength wasshown between PLA with and without HA particles.

The printed trabecular bone showed higher variations in mechanical prop-erties compared to the bulk materials, but higher pullout load than a com-monly used synthetic model when scaled 4.3-fold in printing.

Though further work is needed to increase resolution, especially whenprinting with composite filaments, synthetic models mimicking trabecularstructure with PLA/HA composites -both components are biocompatible andresorbable- using FDM is a promising strategy to develop trabecular modelsfor mechanical evaluation, or potentially clinical applications.

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6. Summary and conclusions

The high prevalence of bone fractures affects the quality of life of manypatients. Complications appear in the surgical treatment of bone fractureswith or without the use of bone substitutes or implants. Due to the fact thatmechanics is highly involved in the locomotion and that bone remodelingresponds to mechanical loading, the mechanical analyses of bone and itsinteraction with implants have to be better understood. The trabecular bonewhich is characterized by its spongy structure is a focus here. As a powerfultool to visualize and quantify the micro-level structural heterogeneity of thetrabecular bone, micro-CT has provided a base for mechanical related anal-yses.

This thesis has discussed several aspects on the topic of mechanical anal-yses of trabecular bone and its interaction with implants based on the en-closed papers. Brief summaries and main conclusions of these papers are listed below:

(1) The currently applied approaches to estimate the elastic modulus oftrabecular bone were reviewed, with highlights on important considerationsof these approaches, and values reported in the literature were summarized.

(2) A global DVC technique was proposed to estimate displacement andstrain fields based on 3D images. The displacement field was discretizedusing higher-order finite elements and solved by a global optimization pro-cedure. Compared to a commercial code, the proposed technique performedsomewhat better in most cases, showing a potential to estimate the displace-ment field with accuracy. This study also highlighted the need for codes tobe applied to double scans, virtual deformation and real deformation testcases when evaluating their performance.

(3) A method was developed to estimate the elastic modulus of single tra-beculae from compression testing. A global DVC algorithm was applied to estimate full-field displacements of the single trabeculae, based on high-resolution 3D images obtained by SRμCT. The displacement provided boundary conditions for high-resolution finite element models considering the irregular geometry of the trabeculae. The proposed method shows a po-tential to estimate single-trabeculae-level mechanical properties, while some possible method improvements were also identified.

(4) Strain fields and cracks of trabecular bone loaded to failure by step-wise compression were estimated by applying DVC on SRµCT images ac-quired during loading. Accumulation of strains and cracks at the individual

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trabecular level were characterized. Trabeculae that were subsequently cracked showed higher strains than regions without segmented cracks during the loading by superimposing the cracks with strain maps. A potential for quantifying individual trabecular deformation by DVC on high-resolution im-ages was shown with this case study.

(5) Screw pullout resistance from trabecular bone with or without aug-mentation with a brushite cement, or a ceramic adhesive was compared. Theenhanced pullout resistance after augmentation with the adhesive, specifical-ly, higher pullout loads, stiffness, fracture energy and more distributedcracks, compared to the cement, indicated a promising augmentation materi-al for the primary implant stability. Cracks at the peripheries of the bone-adhesive composite indicate an observable connection between the threecomponents.

(6) Synthetic trabecular structures were printed using FDM and PLA/HA composite materials. The reproducibility in terms of morphology and increased pullout strength of the printed model indicate a potential for achieving better synthetic models compared to commonly used such, although resolu-tion needs to be improved.

The findings in this thesis improve and may contribute to further im-provements in the mechanical aspects of trabecular bone and implants, in-cluding but not limited to, mechanical properties of bone, augmentationstrategies and synthetic trabecular models for mechanical evaluations.

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7. Future perspectives

Quantitative tomography, imaging analyses and numerical analyses are toolswidely used in tissue related studies; some have even been applied in clinicsfor decades. A combination of these tools with mechanical tests shows agreat potential to analyze the mechanical behavior of materials, especiallyheterogeneous such. Analysis of tomographic images can allow for visuali-zation and quantification of internal deformations. Results from numericalmodels can be compared with image analyses to improve the models or es-timate material mechanical properties. It is expected that these tools will beapplied for the analyses of a wider range of materials, and continuous im-provements in the accuracy or efficiency of these techniques are of greatbenefit. Regarding the topics of this thesis, further investigations or im-provements can be performed, for instance:

· improve the efficiency of the code by programming with a lower-level language;

· a reliable finite element model of screw pullout from trabecular bonemay be proposed by numerical modelling with boundary conditionsestimated by DVC and evaluation of the results based on DVCstrains and the experimentally observed cracks;

· the effect of bonding strength at the interfaces between screwthreads and the augmentation materials on the pullout resistance maybe further investigated by finite element analyses.

· evaluation of strain concentrations further away from the appliedload in the trabecular bone, and quantification of the distribution ofstrain concentrations and crack volume in the bone.

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Svensk sammanfattning

Benfrakturer påverkar många patienters livskvalitet. Komplikationer upp-kommer ofta i samband med frakturrelaterad kirurgisk behandling, både medoch utan benersättningsmaterial eller implantat. Genom att vi rör oss påver-kas ben mekaniskt, och mekanisk last spelar en stor roll för benets remodel-lering. Det är därför viktigt att öka kunskapen om mekaniska aspekter avben, och benets interaktion med implantat, speciellt relaterat till porös ben-struktur (trabekulärt ben). Den här avhandlingen syftar till att undersökatrabekulärt bens mekaniska egenskaper genom analys av isolerade trabekler,hur benersättningsmaterial påverkar implantats primära stabilitet, och till attframställa syntetiska trabekulära modeller för experimentell mekanisk utvär-dering.

En litteraturstudie genomfördes för att ta reda på hur elasticitetsmodulen itrabekulärt ben på vävnadsnivå har undersökts tidigare. I studien identifiera-des fördelar och begränsningar med existerande metoder (Artikel I).

Artikel II utvecklade en global DVC-metod, baserad på diskretiseringmed högre ordningens finita element, för att uppskatta deformation i 3D-bilder. Den utvecklade metoden jämfördes med en annan global DVC-metodgenom både fysisk och virtuell deformation av bilder på trabekulärt ben medolika upplösning. I de flesta fallen presterade den nya tekniken bättre, och atttesta DVC-metoder med flertalet deformationsfält visades vara betydelse-fullt.

I Artikel III utfördes mekaniska tester in-situ på isolerade trabekler frånmänskligt ben under kompression i en SRµCT. Vidare utvecklades en metoddär DVC och FE analys användes för att beräkna elasticitetsmodulen. DVCutfördes på de högupplösta SRµCT-bilderna, och det resulterande deformat-ionsfältet kunde användas som randvillkor till FE modellen av isoleradetrabekler. Metoden visade sig ha potential för utvärdering av mekaniskaegenskaper av isolerade trabekler trots att några potentiella förbättringsmöj-ligheter identifierades.

I Artikel IV belastades trabekulärt ben till brott genom stegvis belastning ikompression. Deformationsfält och sprickor studerades på trabekulär nivågenom att applicera DVC på SRµCT-bilder tagna under belastning. En jäm-förelse mellan deformationsfälten och sprickorna från bilderna visade atttrabekler som gick sönder hade högre deformation i tidigare belastningsstegän regioner utan sprickor. Tekniken som presenteras i denna studie visar

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potential för kvantifiering av deformation i individuella trabekler genomDVC på högupplösta bilder.

Artikel V undersökte hur förstärkning, eller augmentering, av ben medolika material påverkar den primära stabiliteten för ortopediska skruvar itrabekulärt ben. I de mekaniska testerna drogs skruvar ut från mänskligt tra-bekulärt ben, augmenterat med kalciumfosfatbaserat benlim eller kalcium-fosfat-cement, och icke-augmenterat ben användes som kontroll. En del avtesterna utfördes in situ i SRµCT. Ex situ-testerna visade att augmenteringmed benlim gav ett ökat motstånd, och från in situ-bilderna observerades detatt sprickorna var mer jämt distribuerade jämfört med i de andra grupperna.Resultaten visar benlimmets potential för augmentering av skruvar i trabeku-lärt ben.

I Artikel VI utvärderades möjligheten att 3D-printa en trabekulär strukturav PLA/HA-komposit genom FDM-teknik. I studien togs ett första steg föratt framställa syntetiska modeller som efterliknar trabekulär benstruktur,men upplösningen som uppnåddes i studien behöver förbättras.

Med den här avhandlingen förbättras kunskapen om mekaniska aspekter av trabekulärt ben och benets interaktion med implantat. Resultaten bidrar förbättringar i områden relaterade till mekaniska egenskaper av ben, strategier vid augmentering av ortopediska skruvar och framställning av syntetiska benmodeller för experimentell mekanisk utvärdering.

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Acknowledgements

First and foremost, I would like to express my deep and sincere gratitude tomy supervisors, Prof. Cecilia Persson, and Prof. Per Isaksson for giving methe opportunity to carry out studies on this interesting topic. Cecilia, thankyou for your support and encourage. You have given me a chance to collabo-rate with excellent researchers, and exchange knowledge from multidiscipli-nary views. Per, from you, I have learned knowledge in mechanics, skills incoding, and most importantly, independent and critical thinking. Thank youboth for your patient guidance throughout the last four years.

I would like to thank my colleagues in the BMS, MIM and Engineering groups who have helped me. Thank you, Susanne, for sharing opinions in study and life, reminding me of fika as well as helping my Swedish sum-mary. Thanks Caroline for sharing your extensive experience in micro-CT and bone, and always taking time to answer my questions. Thanks Michael and Anna, for the wonderful discussion and help with my experiments. Thanks Jun and David for your help with bone cements. Thanks Fengzhen, Shaohui and Bengt for the discussion in mechanics. Thanks Alejandro and Joakim for your technical support in experimental setups. Thanks Jiaojiao and Rui for your help in porosity test. Thanks Xingxing for welding my electronic boards. Thanks Susan for checking my English writing. Thanks Wei, Le, Charlotte, Luimar, Gry, Céline, Lee, Camilla, Amina, Torbjörn, Marjamy, Yang and everyone in the group for making the working environment so lovely.

A special thank you to my co-authors and collaborators: Stephen Fergu-son for providing human bone, insights and valuable comments in my pro-jects and writing; Thomas Joffre for your support in in-situ experiments and DVC; Nico van Dijk for your amazing work and deep thoughts in DVC; Alicja Bojan and Philip Procter for your endless enthusiasm in the adhesive and screws that made the ex-situ pullout test possible; Anders Palmquist for sharing knowledge in micro-CT and helping me out with morphometric analysis; Andrea Spanou for your hard work in 3D-printing. Thanks to the Paul Scherrer Institute for the provision of synchrotron beamtime, and the beamline scientists, Anne Bonnin, Alessandra Patera, Federica Marone and Gordan Mikuljan, for their valuable help throughout the experiments. A heartful thank you to the ‘synchrotron squad’: Per, Cecilia, Caroline, Thomas, Sara G., Michael, Jenny, Susanne, Anna A.. Thank you for supporting each other and being problem solvers in challenging situations.

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I also want to thank many friends I have made at Uppsala for making life much easier. Bengt and Pranee for introducing the city to me and your kind-hearted suggestions in every aspect. Ivón, Elias, Jinxing&Yingying, Alexey, Juan José, Gabriella, for the memorable afterwork hours. Yuanyuan, my roommate, and Zhen, thank you for your accompany during the years, good and hard times we have spent together, as well as your help with my study. Jun, again, for sharing life experience and nice food with me. The interesting talks and happy times we have during the numerous meals with your guys: Liyang, Bo Cao, Ling, Zhicheng, Dou, Tianbo, etc.

Last but not least, my sincere gratitude to my parents and brother.

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